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\begin{document}

	\title[Existence of solution for Schr\"odinger equation with discontinuous nonlinearity]
	{Existence of solution for Schr\"odinger equation with discontinuous nonlinearity and critical Growth}

	\author{Geovany F. Patricio}

	\address[Geovany F. Patricio]
	{\newline\indent Unidade Acad\^emica de Matem\'atica
		\newline\indent
		Universidade Federal de Campina Grande,
		\newline\indent
		58429-970, Campina Grande - PB - Brazil}
	\email{\href{fernandes.geovany@yahoo.com.br}{fernandes.geovany@yahoo.com.br}}

	\pretolerance10000

	\begin{abstract}
		\noindent This paper concerns with the existence of nontrivial solution for the following problem
		\begin{equation}
		\left\{\begin{aligned}
		-\Delta u + V(x)u & = \gamma H_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber \\
		u \in H^{1}(\mathbb{R}^{N}).
		\end{aligned}
		\right.
		\end{equation}
where, $N\geq 3$, $\gamma\geq 0$, $H_{e}:\mathbb{R} \to \mathbb{R}$ denotes the Heaviside function,
$a \geq 0$, $2<q<2^*$ and  $V:\mathbb{R^{N}} \to \mathbb{R}$ is $\mathbb{Z}^{N}$-periodic with $\beta= 0$ does not belong to the spectrum of $-\Delta+V$.
\end{abstract}

	\thanks{Geovany F. Patricio was
		supported by  CAPES, Brazil }

	\subjclass[2019]{ Primary: 35J15, 35J20, 35A15, 35B33; Secondary: 26A27}
	\keywords{Elliptic equations, Variational methods,  Discontinuous nonlinearity,  Critical growth}

		\maketitle
\section{Introduction}

In this paper we study the existence of nontrivial solution for the following class of elliptic problems
$$
\left\{\begin{aligned}
-\Delta u + V(x)u &= f_{\gamma}(x,u)\;\;\mbox{in}\;\;\mathbb{R}^{N} \\
u \in H^{1}(\mathbb{R}^{N}),
\end{aligned}
\right. \leqno{(P_{\gamma}^{*})}
$$
where, $f_{\gamma}(x,s)=\gamma H_{e}(|s|-a)|s|^{q-2}s+|s|^{2^{*}-2}s$
and $H_e:\mathbb{R} \to \mathbb{R}$ denotes the Heaviside function, that is,
$$
H_e(t)=
\left\{
\begin{array}{l}
	0, \quad \mbox{if} \quad t \leq 0\\
	1, \quad \mbox{if} \quad t > 0.
\end{array}
\right.
$$
In addition, $\gamma\geq 0$, $ N \geq 3$ and $V$ is continuou function, periodic with respect to $x$-variable,
$$
0 \notin \sigma(-\Delta+V),\;\;\mbox{the spectrum of}\;\; -\Delta +V; \leqno{(V_{1})}
$$
and
$$
 \sigma(-\Delta+V)\cap (-\infty, 0)\neq \emptyset. \leqno{(V_{2})}
$$

Problems of type $(P_{\gamma}^{*})$, when nonlinearity is considered continuous, has been studied by Chabrowski and Szulkin \cite{Chabrowski}, M. Schechter and W. Zou \cite{Schechter}, Alves and Germano \cite{AG1}. In \cite{Chabrowski} Chabrowski and Szulkin studied the following class of problems
$$
\left\{\begin{aligned}
-\Delta u + V(x)u &= K(x) |u|^{2^{*}-2}u+g(x,u) \;\;\mbox{in}\;\;\mathbb{R}^{N} \\
u \in H^{1}(\mathbb{R}^{N}),
\end{aligned}
\right.\leqno{(P)}
$$
where $N\geq 4$, $V$, $K$, $g$ are periodic in $x_{j}$ for $1 \leq  j \leq N$, $K > 0$, $g$ is of subcritical growth and $0$ is in a gap of the spectrum of $-\Delta+V$. They proved the existence of a nontrivial solution using the linking theorem.

Knowing that the equation
\begin{equation}\label{Crit}
	-\Delta u+\beta u= |u|^{2^{*}-2}u,\;\mbox{in}\;\;\mathbb{R}^{N},
\end{equation}
when $\beta \neq 0$, has only the trivial solution in $H^{1}(\mathbb{R}^{N})$(cf \cite{Benci}). Thereby, when $\gamma=0$ in $(P_{\gamma}^{*})$ we are left with an equation similar to (\ref{Crit}) and so the existence of nontrivial solution of $(P_{\gamma}^{*})$ is an interesting problem.
One of the pioneering results involving critical problems was obtained in article of Br\'ezis and Nirenberg \cite{Nirenberg} and has helped and motivated a great amount of research on this class of problem involving critical exponent.

In several articles, the authors are focus on equations with subcritical growth, since critical growth bring many difficulties, because, yet in bounded domain $\Omega \subset \mathbb{R}^{N}$, the Sobolev embedding  $H_{0}^{1}(\Omega)\hookrightarrow L^{2^{*}}(\Omega)$ is not compact. In recent years, the study of equations with critical growth has made great progress and has attracted the attention of many authors. One way to regain compactness is constrain the functional value in a suitable interval, see for example \cite{Chabrowski}. When the functional is strongly defined, then the functional has mountain pass geometry, so it is easy to estimate the functional level. For example, under some weaker conditions, Lins and Silva \cite{Lins} considered the existence of nontrivial solutions
\begin{equation*}
-\Delta u+V(x)u=f(x,u)
\end{equation*}
 when $f$ is asymptotically periodic in $x$. However, for the strongly indefinite case, the problem becomes very difficult e.g. see \cite{Chabrowski, Schechter, AG1}.

We will study a class of semilinear problems with discontinuous linearity in $\mathbb{R}^{N}$, strongly indefinite and critical growth $(P_{\gamma}^{*})$. The interest in the study of nonlinear partial differential equations with discontinuous nonlinearities has increased because many free boundary problems arising in mathematical physics may be stated in this from. Among these problems, we have the seepage surface problem and the Elenbaas equation, see for example \cite{Chang1,chang2, chang3}.

Variational methods for problems with discontinuous nonlinearity and critical exponent have been applied to several problems.  Alves and Bertone \cite{alves3}, got two nonnegative solutions for the following quasilinear problem
\begin{equation*}
-\Delta_{p}u = H_{e}(u-a)u^{p^{*}-1}+\lambda h(x),\;\mbox{in}\;\mathbb{R}^{N}
\end{equation*}
where $H_{e}$ is the Heaviside function. Alves, Bertone and Goncalves in \cite{alves1} studied the equation
\begin{equation*}
-\Delta u= u^{2^{*}-1}+\lambda h(x)H(u-a)u^{q},\;\mbox{in}\;\mathbb{R}^{N}
\end{equation*}
obtained the existence of positive solutions for $0\leq q\leq 1$ and $1<q<2^{*}-1.$

A rich literature is available for problems with discontinuous nonlinearities, and we refer the reader to Chang \cite{Chang1}, Ambrosetti and Badiale \cite{Badiale}, Alves and Patricio \cite{AlvesPatricio}, Cerami \cite{cerami}, Alves et al. \cite{alves1}, Alves et al. \cite{alves2}, Alves and Bertone \cite{alves3}, Alves and Nascimento \cite{alves4}, Cerami \cite{cerami, Cerami}, Badiale \cite{badiale2}, Dinu \cite{dinu}, Gasi\'nski and Papageorgiou \cite{GP},  Kourogenis and Papageorgiou \cite{Papageorgiou}, Mironescu and R\u adulescu \cite{Radulescu 2}, R\u adulescu \cite{Radulescu, Radulescu 1, Radulescu 4, Radulescu 5, Radulescu 6, Radulesco 7} and their references. Several techniques have been developed or applied in their study, such as variational methods for nondifferentiable functionals, lower and upper solutions, global branching, and the theory of multivalued mappings.

In this paper a study is made of a strongly indefinite problem with discontinuous nonlinearity that involves two extreme conditions that result in loss of compactness, namely, when we consider the space $\mathbb{R}^{N}$ and the critical growth. More precisely, we will find nontrivial solution to problem $(P_{\gamma}^{*})$ and for this we will use a link theorem for a class of functional locally Lipschitz due to Alves and Patricio \cite{AG}, in which they propose a generalization of the link terorema due to Kryszewski and Szulkin \cite{Kryszewski} in which they consider class $C^{1}$ functional. One of the main difficulty in the present paper was to make sure that the weak limit of sequences $(PS)$ is nontrivial for a given level.

The present paper is relevant, because it is the first result that establishes the existence of solutions for Schr\"odinger equation strongly indefinite with discontinuous nonlinearity and critical growth.

The main result is the following
\begin{theorem}\label{Teorema1}
Suppose that conditions $(V_{1})-(V_{2})$ are satisfied. If $N \geq 4$, the problem $(P_{\gamma}^{*})$ has a nontrivial solution. If $N=3$, there are $\gamma>0$ and $a_{0}=a_{0}(\gamma)>0$ such that, $0\leq a \leq a_{0}$, the problem $(P_{\gamma}^{*})$ has a nontrivial solution.
\end{theorem}
\begin{definition}
For a solution to the problem  $(P_{\gamma}^{*})$
we understand it to be a function $u \in H^{1}(\mathbb{R^{N}})\cap W_{loc}^{2,p}(\mathbb{R^{N}})$, for some $p>1$, such that
$$
-\Delta u(x) + V(x) u(x) \in \partial_{t} F_{\gamma}(x,u(x))\;\;\mbox{a.e in}\;\; \mathbb{R}^{N},
$$
where $F_{\gamma}(x,t)= \int_{0}^{t}f_{\gamma}(x,s)ds$ and $\partial_t F_{\gamma}$ denotes the generalized gradient of $F_{\gamma}$ with respect to variable $t$.
When
\begin{equation*}
-\Delta u(x)+V(x)u(x)=f_{\gamma}(x,u(x)),\;\mbox{a.e in}\;\mathbb{R}^{N},
\end{equation*}
then we say that $u$ is strong solution.
\end{definition}
\begin{remark}\label{Obs1}
In case $N\geq 4$, we will not have restrictions for constants $a\geq 0$ and $\gamma\geq 0$ to obtain a nontrivial solution to the problem $(P_{\gamma}^{*})$. Thereby, in case $N\geq 4$, we will have a strong solution for $(P_{\gamma}^{*})$
choosing $a>0$ so that
\begin{equation}\label{V}
\sup_{x \in \mathbb{R}^{N}}V(x)<a^{2^{*}-2}.
\end{equation}
First note that (\ref{V}) is possible because we are assuming that $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is $\mathbb{Z}^{N}$-periodic and therefore bounded, i.e, there exists $K_{0}>0$ such that $|V(x)|\leq K_{0}$ for all $x \in \mathbb{R}^{N}$. So, just choose $a>K_{0}$.

Using the definition of generalized gradient, by a simple calculation, we get:
$$
\partial_{t}F_{\gamma}(x,u(x))= \left\{\begin{aligned}
u(x)^{2^{*}-1},\;\;\mbox{if}\;\; 0\leq u(x)<a \\
[a^{2^{*}-1}, a^{2^{*}-1}+\gamma a^{q-1}],\;\; \mbox{if}\;\; u(x)=a \\
u(x)^{2^{*}-1}+\gamma u(x)^{q-1},\;\; \mbox{if}\;\; u(x)>a.
\end{aligned}
\right.
$$
or
$$
\partial_{t}F_{\gamma}(x,u(x))= \left\{\begin{aligned}
(-u(x))^{2^{*}-2}u(x),\;\;\mbox{if}\;\; -a<u(x)\leq 0 \\
[ (-a)^{2^{*}-1}+\gamma (-a)^{q-1}, (-a)^{2^{*}-1}],\;\; \mbox{if}\;\; u(x)=-a \\
(-u(x))^{2^{*}-2}u(x)+\gamma (-u(x))^{q-2}u(x),\;\; \mbox{if}\;\; u(x)<-a.
\end{aligned}
\right.
$$
\end{remark}
Suposse that $u$ is a solution of $(P_{\gamma}^{*})$, then
$$
 \left\{\begin{aligned}
-\Delta u(x)+V(x)u(x)=u(x)^{2^{*}-1},\;\;\mbox{if}\;\; 0\leq u(x)<a \\
-\Delta u(x)+V(x)u(x) \in [a^{2^{*}-1}, a^{2^{*}-1}+\gamma a^{q-1}],\;\; \mbox{if}\;\; u(x)=a \\
-\Delta u(x)+V(x)u(x)=u(x)^{2^{*}-1}+\gamma u(x)^{q-1},\;\; \mbox{if}\;\; u(x)>a.
\end{aligned}
\right.
$$
or
$$
\left\{\begin{aligned}
-\Delta u(x)+V(x)u(x)=(-u(x))^{2^{*}-2}u(x),\;\;\mbox{if}\;\; -a<u(x)\leq 0 \\
-\Delta u(x)+V(x)u(x) \in [ (-a)^{2^{*}-1}+\gamma (-a)^{q-1}, (-a)^{2^{*}-1}],\;\; \mbox{if}\;\; u(x)=-a \\
-\Delta u(x)+V(x)u(x)=(-u(x))^{2^{*}-2}u(x)+\gamma (-u(x))^{q-2}u(x),\;\; \mbox{if}\;\; u(x)<-a.
\end{aligned}
\right.
$$
Set
\begin{equation*}
A_{+}=\{x \in \mathbb{R}^{N}: u(x)=a\}\;\;\mbox{and}\;\;A_{-}=\{x \in \mathbb{R}^{N}: u(x)=-a\}.
\end{equation*}
Under (\ref{V}) we obtain that the Lebesgue
measure of $A_{+}$ (or $A_{-}$) is zero. Otherwise if measure of $A_{+}$ (or  $A_{-}$) is positive, then by applying Stampacchia theorem \cite{Stampacchia},
\begin{equation*}
-\Delta u(x)=0\;\;\mbox{in}\;\;  A_{+}\; (\mbox{or}\; A_{-}),
\end{equation*}
that is,
\begin{eqnarray}
&&V(x)a \in [a^{2^{*}-1}, a^{2^{*}-1}+\gamma a^{q-1}],\;\mbox{for}\;\; x \in A_{+}, \nonumber \\
\mbox{or}\nonumber \\
&&-V(x)a \in [ (-a)^{2^{*}-1}+\gamma (-a)^{q-1}, (-a)^{2^{*}-1}],\;\mbox{for}\;\; x \in A_{-}. \nonumber
\end{eqnarray}
In any of the above cases, once $(-a)^{2^{*}-2}= a^{2^{*}-2}$, we have
\begin{equation*}
a^{2^{*}-2}\leq V(x),\;\mbox{for all}\;\;x \in A_{+}\;(\mbox{or}\;A_{-}),
\end{equation*}
a contradiction with (\ref{V}).

By remark \ref{Obs1}, we can enunciate the following:
\begin{corollary}
	If $N\geq 4$, assuming $(V_{1})$ and $(V_{2})$, there exist $a>0$ such that the problem $(P_{\gamma}^{*})$ has a nontrivial strong solution.
\end{corollary}

\noindent \textbf{Notation:} From now on, otherwise mentioned, we use the following notations:
\begin{itemize}
	\item $B_r(u)$ is an open ball centered at $u$ with radius $r>0$, $B_r=B_r(0)$.

	\item $X^{*}$ denotes the dual topological space of $X$ and $||\;\;\;||_{*}$ denotes the norm in $X^{*}$.

	\item   $C$ denotes any positive constant, whose value is not relevant.

	\item  $||\,\,\,||_p$ denotes the usual norm of the Lebesgue space $L^{p}(\mathbb{R}^N)$, for $p \in [1,+\infty]$.

	\item  $||\,\,\,||_{H}$ denotes the usual norm of the Orlicz space $L^{H}(\mathbb{R}^N)$ associated the $N$-function $H$.

	\item If $u:\mathbb{R}^N \to \mathbb{R}$ is mensurable function, the integral $\int_{\mathbb{R}^N}u\,dx$ will be denoted by $\int_{\mathbb{R}^N}u$.

	\item $o_{n}(1)$ denotes a real sequence with $o_{n}(1)\to 0$ as $n \to +\infty$.

	\item If $k>0$, $O(\varepsilon^{k})$ denotes a function that $\frac{O(\varepsilon^{k})}{\varepsilon^{k}}$ is bounded as $\varepsilon\rightarrow 0^{+}$.

\end{itemize}

\section{Basic results from nonsmooth analysis}

In this section, for the reader's convenience, we recall some definitions and basic results on the critical point theory of locally Lipschitz functionals as developed by Chang \cite{Chang1}, Clarke \cite{Clarke, Clarke1} and Grossinho and Tersian \cite{rosario}.

Let $(X, ||\;||_{X})$ and $(Y, ||\;||_{Y})$ be a real Banach spaces. A functional $I :X \rightarrow \mathbb{R} $ is locally Lipschitz, $I\in Lip_{loc}(X, \mathbb{R})$ for short, if given $u \in X$ there is an open neighborhood $V:=V_{u} \subset X$ of $u$, and a constant $K=K_{u}>0$ such that
\begin{equation*}
	|I(v_{2})-I(v_{1})| \leq K ||v_{1}-v_{2}||_{X},\;\;v_{i} \in V,\;i=1,2.
\end{equation*}
The generalized directional derivative of $I$ at $u$ in the direction of $v \in X$ is defined by
\begin{equation*}
	I^{\circ}(u;v)= \limsup_{h\rightarrow 0, \delta \downarrow 0}\frac{1}{\delta}\left(I(u+h+\delta v)-I(u+h)\right).
\end{equation*}
The generalized gradient of $I$ at $u$ is the set
\begin{equation*}
	\partial I(u)= \{\xi \in X^{*}\;;\; I^{\circ}(u;v) \geq \left<\xi, v\right>\,;\forall\; v \in X \}.
\end{equation*}
\begin{lemma}
	If $I$ is continuously differentiable to Fr\'echet in an open neighborhood of $u \in X$, we have $\partial I(u)= \{I'(u)\}$.
\end{lemma}
\begin{lemma}\label{35}
	If $Q \in C^{1}(X, \mathbb{R})$ and $\Psi \in Lip_{loc}(X, \mathbb{R})$, then for each $u \in X$
	$$
	\partial(Q+\Psi)(u)= Q'(u) + \partial \Psi(u).
	$$
\end{lemma}
Moreover, we denote by $\lambda_{I}(u)$ the following real number
$$
\lambda_{I}(u):=\min\{||\xi||_{*}: \xi \in \partial I(u)\}.
$$
We recall that $u \in X$ is a critical point of $I$ if $0 \in \partial I(u)$, or equivalently, when $\lambda_I(u)=0$.

\section{On the energy functional of problem $(P_{\gamma}^{*})$}

It follows  that the functional
$$I_{\gamma}(u)= \frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)u^{2})-\int_{\mathbb{R}^{N}}F_{\gamma}(x,u), \;u \in H^{1}(\mathbb{R}^{N}),$$
where $F_{\gamma}(x,t)= \int_{0}^{t}f_{\gamma}(x,s)ds$, is well defined.

By standard argument, $Q \in C^{1}(H^{1}(\mathbb{R}^{N}), \mathbb{R})$
where
$$ Q(u)= \frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)u^{2})$$
and
$$Q'(u)v=\int_{\mathbb{R}^{N}}(\nabla u \nabla v+V(x)u v),\;\forall\; u,v \in H^{1}(\mathbb{R}^{N}).$$
In addition, we can write
$$I_{\gamma}= Q- \Psi_{\gamma},$$
where $\Psi_{\gamma}: H^{1}(\mathbb{R}^{N})\rightarrow \mathbb{R}$ given by
\begin{equation*}
\Psi_{\gamma}(u)=\int_{\mathbb{R}^{N}}F_{\gamma}(x,u).
\end{equation*}
By $(V_{1})$, it is well known that $H^{1}(\mathbb{R}^{N})=X^{+}\oplus X^{-}$ is a orthogonal decomposition and there is an equivalent norm $||\cdot||$ to $||\cdot||_{H^{1}(\mathbb{R}^{N})}$ (see \cite{AG, AlvesPatricio}) such that
\begin{equation}\label{Functional}
I_{\gamma}(u)= \frac{1}{2}||u^{+}||^{2}-\frac{1}{2}||u^{-}||^{2}-\Psi_{\gamma}(u),\;\forall\; u=u^{+}+u^{-} \in X^{+} \oplus X^{-} .
\end{equation}
By definition
\begin{equation}\label{Grad.}
\partial_{t} F_{\gamma}(x,t)= \{\mu \in \mathbb{R}\;:\; F_{\gamma}^{\circ}(x,t;r) \geq \mu r,\; r \in \mathbb{R}\},
\end{equation}
where $F_{\gamma}^{\circ}(x,t;r)$ denotes the generalized directional derivative of $t\mapsto F_{\gamma}(x,t)$ in the direction
of $r$, i.e,
\begin{equation*}
F_{\gamma}^{\circ}(x,t;r)= \limsup_{h\rightarrow t, \lambda \downarrow 0} \frac{F_{\gamma}(x, h+\lambda r)-F_{\gamma}(x,h)}{\lambda}.
\end{equation*}
Consider
\begin{eqnarray}
F_{\gamma}:\mathbb{R}^{N}&\times& \mathbb{R}\longrightarrow \mathbb{R} \nonumber \\
&(x,t)&\longmapsto F_{\gamma}(x,t) = \int_{0}^{t}f_{\gamma}(x,s) ds \nonumber
\end{eqnarray}
where $f_{\gamma}(x,s)= \gamma H(|s|-a)|s|^{q-2}s+|s|^{2^{*}-2}s$, that is,
$$
F_{\gamma}(x,t)= \left\{\begin{aligned}
\frac{1}{2^{*}}|t|^{2^{*}},\;\;\mbox{if}\;\; |t|\leq a \\
\frac{1}{2^{*}}|t|^{2^{*}}+\frac{\gamma}{q}|t|^{q}- \frac{\gamma}{q}a^{q},\;\; \mbox{if}\;\; |t|>a
\end{aligned}
\right.
$$
In fact it is easy to check that functional $\Psi_{\gamma}: H^{1}(\mathbb{R}^{N})\rightarrow \mathbb{R}$ given by
\begin{equation} \label{Psi}
\Psi_{\gamma}(u)=\int_{\mathbb{R}^{N}}F_{\gamma}(x,u),
\end{equation}
is well defined. However, in order to apply variational methods it is better to consider the functional $\Psi$ in a more appropriated domain, that is, $\Psi_{\gamma}: L^{\Phi}(\mathbb{R}^{N})\rightarrow \mathbb{R}$, for $\Phi(t)=|t|^{q}+|t|^{2^{*}}$, where $ L^{\Phi}(\mathbb{R}^{N})$ denotes the Orlicz space associated with the $N$-function $\Phi$ (for more details on Orlicz space see \cite{Fukagai 1, RAO, T.K}).

Since $\Phi$ satisfies $\Delta_{2}$-condition, we can guarantee that given $J \in (L^{\Phi}(\mathbb{R}^{N}))^{*}$, then
$$J(u)= \int_{\mathbb{R}^{N}} vu\;,\;\;\forall\; u \in L^{\Phi}(\mathbb{R}^{N}),$$
for some $v \in L^{\tilde{\Phi}}(\mathbb{R}^{N})$, where $\tilde{\Phi}$ is the conjugate function of $\Phi$. In general, we need to prove that the inclusion below holds
$$\partial \Psi_{\gamma} (u)\subset \partial_{t} F_{\gamma}(x,u)= [\underline{f_{\gamma}}(x, u(x)), \overline{f_{\gamma}}(x, u(x))]\;\;\mbox{a.e in}\;\; \mathbb{R}^{N},$$
where
$$\underline{f_{\gamma}}(x,t)= \lim_{r\downarrow 0} ess \inf\{f_{\gamma}(x,s); |s-t|<r\}$$
and
$$\overline{f_{\gamma}}(x,t)= \lim_{r\downarrow 0} ess \sup\{f_{\gamma}(x,s); |s-t|<r\}.$$
We have that the condition below is satisfied:
\begin{itemize}
	\item [($F_{*}$)]There exist $C_0, C_1>0$ such that
	$$|\xi| \leq C_0(|u|^{q-1} +|u|^{2^{*}-1}) \leq C_1\Phi'(|u|), \;\;\forall\; \xi \in \partial_{t} F_{\gamma}(x,u),\;\forall\; x \in \mathbb{R}^{N},$$
	for some $C_1>0$.
\end{itemize}

The next three results establish important properties of the functional $\Psi_{\gamma}$ given in (\ref{Psi}).

\begin{lemma}\label{Lip}[See \cite{Alves1} or \cite{AG1}]
	Assume ($F_{*}$). Then, the functional $\Psi_{\gamma}: L^{\Phi}(\mathbb{R}^{N})\rightarrow \mathbb{R}$ given by
	$$\Psi_{\gamma}(u)= \int_{\mathbb{R}^{N}} F_{\gamma}(x,u),\; u \in L^{\Phi}(\mathbb{R}^{N}),$$
	is well defined and $\Psi_{\gamma} \in Lip_{loc}(L^{\Phi}(\mathbb{R}^{N}), \mathbb{R})$.
\end{lemma}

\begin{theorem}[See \cite{Chang1}, Theorem 2.1  or \cite{Alves}, Theorem 4.1]\label{72}
	Assume ($F_{*}$), then for each $u \in L^{\Phi}(\mathbb{R}^{N})$,
	\begin{equation}\label{67}
	\partial \Psi_{\gamma} (u) \subset \partial_{t} F_{\gamma}(x, u) = [\underline{f_{\gamma}}(x,u(x)), \overline{f_{\gamma}}(x,u(x))]\;\;\mbox{a.e in}\;\; \mathbb{R}^{N}.
	\end{equation}
\end{theorem}
The inclusion above means that given $\xi \in  \partial \Psi_{\gamma} (u) \subset (L^{\Phi}(\mathbb{R}^{N}))^{*}\approx L^{\tilde{\Phi}}(\mathbb{R}^{N})$, there is $\tilde{\xi} \in L^{\tilde{\Phi}}(\mathbb{R}^{N})$ such that
\begin{itemize}
	\item $\left<\xi, v \right>= \int_{\mathbb{R}^{N}} \tilde{\xi}v ,\;\forall\; v \in L^{\Phi}(\mathbb{R}^{N})$,
	\item $\tilde{\xi}(x) \in \partial_{t} F_{\gamma}(x, u(x)) = [\underline{f_{\gamma}}(x,u(x)), \overline{f_{\gamma}}(x,u(x))]\;\;\mbox{a.e in}\;\; \mathbb{R}^{N}.$
\end{itemize}
The following proposition is very important to establish the existence of a critical point for the functional $I_{\gamma}$.
\begin{proposition}(See \cite{AG}).\label{7}
	If $(u_{n}) \subset H^{1}(\mathbb{R}^{N})$  is such that
	$u_{n}\rightharpoonup u_{0}$ in $H^{1}(\mathbb{R}^{N})$ and $\rho_{n} \in \partial \Psi_{\gamma}(u_{n})$ satisfies $\rho_{n} \stackrel{\ast}{\rightharpoonup} \rho_{0}$ in $(H^{1}(\mathbb{R}^{N}))^{*}$, then $\rho_{0} \in \partial \Psi_{\gamma}(u_{0})$.
\end{proposition}

\section{Generalized linking theorem}

	From now on, $X$ is a Hilbert space with $X=Y \oplus Z$, where $Y$ is a separable closed subspace of $X$ and $Z=Y^{\perp}$. If $u \in X$,  $u^{+}$ and $u^{-}$ denote the orthogonal projections from $X$ in $Z$ and in $Y$, respectively. In $X$ let us define the norm
	\begin{eqnarray}
	|||\cdot|||: &X &\longrightarrow \mathbb{R} \nonumber \\
	&u& \longmapsto |||u|||=\max\left\{||u^{+}||, \sum_{k=1}^{\infty}\frac{1}{2^{k}}|(u^{-},e_{k})|\right\}, \nonumber
	\end{eqnarray}
	where $(e_{k})$ is a total orthonormal sequence in $Y$. The topology on $X$ generated by $|||\cdot|||$ will be denoted by $\tau$ and all topological notions related to it will include this symbol.

Let $I :X \rightarrow \mathbb{R}$ a funtional locally Lipschitz, $I\in Lip_{loc}(X, \mathbb{R})$. We will say a functional $I:X\rightarrow \mathbb{R}$ verifies the condition $(H)$ when: \\
$$
(H) \quad \left\{
\begin{array}{l}
	\mbox{If $(u_{n}) \subset I^{-1}([\alpha, \beta])$ is such that $u_{n}\stackrel{\tau}{\rightarrow} u_{0}$ in $X$, then there exists $M>0$ }  \\
	\mbox{such that $\partial I(u_{n}) \subset B_{M}(0) \subset X^{*},\;\forall\; n \in \mathbb{N}$. In addition, if  $\xi_{n} \in \partial I(u_{n})$} \\
	\mbox{ with $\xi_{n} \stackrel{\ast}{\rightharpoonup} \xi_{0}$ in  $X^{*}$, we have $\xi_{0} \in \partial I(u_{0})$.}
\end{array}
\right\}
$$
\begin{theorem}\label{Link}(See \cite{AG}).
Let $Y$ be a separable closed subspace of a Hilbert space $X$ and $Z=Y^{\perp}$. If $u \in X$, as in the previous section, $u^{+}$ and $u^{-}$ denote the orthogonal projections in $Z$ and $Y$, respectively.

Given $\rho >r>0$ and $z \in Z$ with $||z||=1$, we set
\begin{eqnarray}
&&\mathcal{M}=\left\{u=y+t z\;; ||u||\leq \rho, t \geq 0 \;\;\mbox{and}\;\; y \in Y\right\} \nonumber \\
&& \mathcal{M}_{0}=\left\{u=y+t z\;;y \in Y, ||u||= \rho\;\;\mbox{and}\;\; t \geq 0 \;\;\mbox{or}\;\;  ||u||\leq  \rho\;\;\mbox{and}\;\; t=0 \right\} \nonumber \\
&& S=\left\{u \in Z\;; ||u||=r\right\}. \nonumber
\end{eqnarray}
Assume $I \in Lip_{loc}(X, \mathbb{R})$ such that
\begin{equation*}
I \;\mbox{is}\;\;\tau-\mbox{upper semicontinuous}
\end{equation*}
and
\begin{equation*}
b=\inf_{S}I> \sup_{\mathcal{M}_{0}}I\;,\; d=\sup_{\mathcal{M}} I < \infty.
\end{equation*}
If $I$ verifies the condition $(H)$, there is $c \in [b,d]$ and a sequence $(u_{n}) \subset X$ such that
$$I(u_{n}) \rightarrow c\;\;\mbox{and}\;\; \lambda_{I}(u_{n})\rightarrow 0.$$
\end{theorem}

\section{Proof of Theorem \ref{Teorema1}}

It follows, by Alves and Patricio \cite{AG}, that the functional $I_{\gamma}$ checks condition $(H)$ and the hypotheses of the Theorem \ref{Link}. Then, there is $c_{\gamma} \in [b_{\gamma},d_{\gamma}]$ and a sequence $(u_{n}) \subset H^{1}(\mathbb{R}^{N})$ bounded (see \cite[lemma 6.9]{AG}) such that
	\begin{equation*}
		I_{\gamma}(u_{n})\rightarrow c_{\gamma}\;\;\mbox{and}\;\; \lambda_{I_{\gamma}}(u_{n})\rightarrow 0.
	\end{equation*}

	\begin{claim}
		There exists $\delta>0$ such that
		$$\liminf_{n} \sup_{y \in \mathbb{R}^{N}} \int_{B(y,1)} |u_{n}|^{2^{*}}\geq \delta.$$
	\end{claim}
	If the claim is not true, we must have
	$$\liminf_{n} \sup_{y \in \mathbb{R}^{N}} \int_{B(y,1)} |u_{n}|^{2^{*}}=0.$$
	Thus, applying \cite [Lemma 2.1]{Ramos},  $u_{n}\rightarrow 0$ in $L^{2^{*}}(\mathbb{R}^{N})$ and
	by interpolation on the Lebesgue spaces, $u_{n}\rightarrow 0$ in $L^{s}(\mathbb{R}^{N})$ for $2<s< 2^{*}$.
	On the other hand,
	\begin{equation}\label{70}
0<c=I_{\gamma}(u_{n})- \frac{1}{2}\left<w_{n}, u_{n} \right>+o_{n}(1)=\left(\frac{1}{2}- \frac{1}{2^{*}}\right) \int_{\mathbb{R}^{N}} \rho_{n} u_{n} +o_{n}(1),
	\end{equation}
	where $w_{n}= Q'(u_{n})- \rho_{n}$ with $\lambda_{I_{\gamma}}(u_{n})=||w_{n}||_{*}$ and $\rho_{n} \in  \partial \Psi_{\gamma}(u_{n})$.

	Since
	$$\int_{\mathbb{R}^{N}} \rho_{n} u_{n} \leq \gamma ||u_{n}||_q^{q}+||u_{n}||_{2^{*}}^{2^{*}}\rightarrow 0,$$
	contrary to (\ref{70}).

	From this, going to a subsequence if necessary, there exists $n_{0} \in \mathbb{N}$ such that
	$$\sup_{y \in \mathbb{R}^{N}} \int_{B(y,1)} |u_{n}|^{2^{*}} \geq \frac{\delta}{2}, \;n \geq n_{0}.$$
	By definition of supreme, there exists $(y_{n}) \subset \mathbb{R}^{N}$ such that
	\begin{equation*}
	\int_{B(y_{n},1)} |u_{n}|^{2^{*}} \geq \frac{\delta}{4}, \;n \geq n_{0}.
	\end{equation*}
	Then, there exists $(z_{n}) \subset \mathbb{Z}^{N}$ such that
	\begin{equation*}
	\int_{B(z_{n},1+\sqrt{N})} |u_{n}|^{2^{*}} \geq \frac{\delta}{4}, \;n \geq n_{0}.
	\end{equation*}
	Setting $v_{n}(x)=u_{n}(x+z_{n})$, we compute
	\begin{equation}\label{84}
	\int_{B(0,1+\sqrt{N})} |v_{n}(x)|^{2^{*}}= \int_{B(z_{n},1+\sqrt{N})} |u_{n}(x)|^{2^{*}} \geq \frac{\delta}{4},\;n \geq n_{0}.
	\end{equation}
Similarly to what was done in \cite[Claim 6.12]{AG}, we have that $(v_{n}) \subset H^{1}(\mathbb{R}^{N})$ is also a $(PS)_{c_{\gamma}}$ sequence for $I_{\gamma}$.
Going to a subsequence, if necessary, let $v \in H^{1}(\mathbb{R}^{N})$ the weak limit of the sequence $(v_{n}) \subset H^{1}(\mathbb{R}^{N})$.

\begin{claim}\label{103}
	If $c_{\gamma}<\dfrac{S^{\frac{N}{2}}}{N}$, then $v\neq 0$.
\end{claim}
Suppose by contradiction $v=0$ and assume that
$$|\nabla v_{n}|^{2} \rightharpoonup \mu\;\;\mbox{and}\;\;|v_{n}|^{2^{*}} \rightharpoonup \nu \;\;\mbox{in }\;\; \mathcal{M}(\mathbb{R}^{N}).$$
By Concentration-Compactness Principle II due to Lions \cite{Principio}, there exist a countable  set $J$, $\nu_{j} \in \mathbb{R}_{+}$ and $(x_{j})_{j} \subset \mathbb{R}^{N}$ such that
$$\nu= \sum_{j \in J} \nu_{j} \delta_{x_{j}}$$
where $\delta_{x}$ denotes the mass of Dirac concentrated in $x \in \mathbb{R}^{N}$. In addition,
$$\mu \geq  S \sum_{j \in J} \nu_{j}^{\frac{2}{2^{*}}} \delta_{x_{j}},$$
where
$$S= \inf \{|\nabla u|_{2}^{2}\;;\; u \in D^{1,2}(\mathbb{R}^{N})\;,\; |u|_{2^{*}}=1\}$$
is the best Sobolev constant for immersion of $D^{1,2}(\mathbb{R}^{N})$ in $L^{2^{*}}(\mathbb{R}^{N})$. We prove that $\nu_{j}=0$ for all $j \in J$. Indeed, otherwise, suppose there is $j_{0} \in J$ such that
		\begin{equation}\label{99}
		\nu_{j_{0}}>0.
		\end{equation}
For $\delta>0$ consider the function $\varphi_{\delta} \in C_{0}^{\infty}(\mathbb{R}^{N})$

		$
		\varphi_{\delta}(x) = \left\{\begin{aligned}
		1,\;\; |x-x_{j_{0}}|\leq \frac{\delta}{2} \\
		0,\;\; |x-x_{j_{0}}|>\delta.
		\end{aligned}
		\right.
		$

		By definition of convergence in the sense of measure theory, we get
		\begin{itemize}
			\item $\displaystyle\int_{\mathbb{R}^{N}} \varphi_{\delta} |v_{n}|^{2^*}\rightarrow  \int_{\mathbb{R}^{N}} \varphi_{\delta} d\nu$, as $n\rightarrow +\infty$.
			\item $\displaystyle \int_{\mathbb{R}^{N}} \varphi_{\delta} |\nabla v_{n}|^{2}\rightarrow \int_{\mathbb{R}^{N}} \varphi_{\delta} d\mu$, as $n\rightarrow +\infty$.
		\end{itemize}
		Since $(v_{n})_{n}$ is bounded in $L^{2^{*}}(\mathbb{R}^{N})$, then $(|v_{n}|^{q})_{n}$ is bounded in $L^{\frac{2^{*}}{q}}(\mathbb{R}^{N})$ where $q \in (2, 2^{*})$ and $v_{n}(x)\rightarrow 0$ a.e in $\mathbb{R}^{N}$. So,
		$$|v_{n}|^{q}\rightharpoonup 0\;\; \mbox{in}\;\; L^{\frac{2^{*}}{q}}(\mathbb{R}^{N}),$$
		that is,
		\begin{equation}\label{78}
		\int_{\mathbb{R}^{N}} |v_{n}|^{q} \varphi_{\delta} \rightarrow 0.
		\end{equation}
		By the inequality of H\"older and the limitation of $(v_{n})_{n}$ in $H^{1}(\mathbb{R}^{N})$, we get
		\begin{eqnarray}
		\left|\int_{\mathbb{R}^{N}} v_{n} \nabla v_{n} \nabla \varphi_{\delta}\right| &\leq& \left(\int_{\mathbb{R}^{N}} |v_{n}|^{2}  |\nabla \varphi_{\delta}|^{2}\right)^{\frac{1}{2}} \left(\int_{\mathbb{R}^{N}}|\nabla v_{n}|^{2}\right)^{\frac{1}{2}} \nonumber \\
		&\leq& C\left(\int_{B_{\delta}(x_{j_{0}})} |v_{n}|^{2}  |\nabla \varphi_{\delta}|^{2}\right)^{\frac{1}{2}}\nonumber.
		\end{eqnarray}
		Once $v_{n}\rightarrow 0$ in $L_{loc}^{2}(\mathbb{R}^{N})$, we obtain
		$$\left(\int_{B_{\delta}(x_{j_{0}})} |v_{n}|^{2}  |\nabla \varphi_{\delta}|^{2}\right)^{\frac{1}{2}}\rightarrow 0,\;\mbox{as}\;\; n\rightarrow \infty,$$
		showing that
		\begin{equation}\label{79}
		\int_{\mathbb{R}^{N}} v_{n} \nabla v_{n} \nabla \varphi_{\delta}\rightarrow 0, \;\mbox{as}\; n\rightarrow +\infty.
		\end{equation}
		In addition,
		\begin{equation}\label{80}
		\int_{\mathbb{R}^{N}} V(x) \varphi_{\delta} |v_{n}|^{2}\rightarrow 0, \;\mbox{as}\; n\rightarrow +\infty.
		\end{equation}
		We still have
		\begin{equation}\label{81}
		\int_{\mathbb{R}^{N}} \rho_{n} v_{n} \varphi_{\delta} \leq \gamma\int_{\mathbb{R}^{N}} \varphi_{\delta} |v_{n}|^{q} + \int_{\mathbb{R}^{N}} \varphi_{\delta} |v_{n}|^{2^{*}}.
		\end{equation}
		By the fact that $(v_{n})$ is $(PS)_{c_{\gamma}}$, there is $w_{n} \in \partial I_{\gamma}(v_{n})$ and $\rho_{n} \in \partial \Psi(v_{n})$ such that
		$$||w_{n}||=\lambda_{I_{\gamma}}(v_{n})=o_{n}(1)\;\mbox{and}\; \left<w_{n}, \phi\right>= \left<Q'(v_{n}), \phi\right>- \left<\rho_{n}, \phi\right>,\;\forall\; \phi \in  H^{1}(\mathbb{R}^{N}).$$
		By (\ref{78}), (\ref{79}), (\ref{80}) and (\ref{81})
		\begin{eqnarray}
		o_{n}(1)=\left<w_{n}, \varphi_{\delta} v_{n} \right> &=& \int_{\mathbb{R}^{N}} \nabla v_{n} \nabla (\varphi_{\delta} v_{n}) + \int_{\mathbb{R}^{N}} V(x) |v_{n}|^{2} \varphi_{\delta} - \int_{\mathbb{R}^{N}} \rho_{n} v_{n} \varphi_{\delta} \nonumber \\
		&=& \int_{\mathbb{R}^{N}} |\nabla v_{n}|^{2} \varphi_{\delta} + \int_{\mathbb{R}^{N}} v_{n} \nabla \varphi_{\delta} \nabla v_{n} +\int_{\mathbb{R}^{N}} V(x) |v_{n}|^{2} \varphi_{\delta} - \int_{\mathbb{R}^{N}} \rho_{n} v_{n} \varphi_{\delta} \nonumber \\
		&\geq& \int_{\mathbb{R}^{N}} |\nabla v_{n}|^{2} \varphi_{\delta} + \int_{\mathbb{R}^{N}} v_{n} \nabla \varphi_{\delta} \nabla v_{n} +\int_{\mathbb{R}^{N}} V(x) |v_{n}|^{2} \varphi_{\delta}+\nonumber \\
		&-& \gamma\int_{\mathbb{R}^{N}} \varphi_{\delta} |v_{n}|^{q} - \int_{\mathbb{R}^{N}} \varphi_{\delta} |v_{n}|^{2^{*}}, \nonumber
		\end{eqnarray}
		that is,
		$$0 \geq \int_{\mathbb{R}^{N}} \varphi_{\delta} d\mu - \int_{\mathbb{R}^{N}} \varphi_{\delta} d\nu,\;\forall\;\delta>0.$$
		Crossing the limit when $\delta\rightarrow 0$, by the dominated convergence theorem of Lebesgue, we get the following relationship
		$$\mu(x_{j_{0}}) \leq \nu(x_{j_{0}}),$$
		this is,
		$$S\nu_{j_{0}}^{\frac{2}{2^{*}}}\leq \nu_{j_{0}}.$$
		By (\ref{99})
		\begin{equation}\label{83}
		\nu_{j_{0}} \geq S^{\frac{N}{2}}.
		\end{equation}
		Knowing $\rho_{n}(x) \in \partial_{t} F_{\gamma}(x,v_{n}(x))$
		\begin{eqnarray}
		o_{n}(1)+c_{\gamma}&=& I_{\gamma}(v_{n})- \frac{1}{2}\left<w_{n}, v_{n} \right> \nonumber\\
		&=& \frac{1}{2}\int_{\mathbb{R}^{N}} \rho_{n} v_{n} - \int_{\mathbb{R}^{N}} F_{\gamma}(x,v_{n}) \nonumber \\
		&=& \left(\frac{1}{2}-\frac{1}{2^{*}}\right) \int_{\mathbb{R}^{N}} \rho_{n} v_{n} \geq \frac{1}{N} \int_{\mathbb{R}^{N}} |v_{n}|^{2^{*}}, \nonumber
		\end{eqnarray}
		this is,
		\begin{equation*}
		\frac{1}{N} \liminf_{n}\left(\int_{\mathbb{R}^{N}} |v_{n}|^{2^{*}} \right)\leq c_{\gamma}.
		\end{equation*}
		Since $|v_{n}|^{2^{*}}\rightharpoonup \nu$ in $\mathcal{M}^{+}(\mathbb{R}^{N})$ and (\ref{83}), we get
		\begin{eqnarray}
		c_{\gamma} &\geq& \frac{1}{N} \liminf_{n} \left(\int_{\mathbb{R}^{N}} |v_{n}|^{2^{*}} \right)\nonumber  \\
		&\geq & \frac{1}{N}\nu(\mathbb{R}^{N}) \nonumber \\
		& \geq & \frac{1}{N} \nu(\left\{x_{j_{0}}\right\}) = \frac{1}{N} \nu_{j_{0}} \geq \frac{1}{N} S^{\frac{N}{2}}>c_{\gamma}, \nonumber
		\end{eqnarray}
		what is absurd.

		Therefore, $\nu=0$ implying in $|v_{n}|^{2^{*}}\rightharpoonup 0$ in $\mathcal{M}^{+}(\mathbb{R}^{N})$ and consequently $v_{n}\rightarrow 0$ in $L_{loc}^{2^{*}}(\mathbb{R}^{N})$ contradicting (\ref{84}).

Now, we are ready to show the estimate from above involving the number $c_{\gamma}>0$.
\begin{itemize}
	\item Case $N\geq 4$.
\end{itemize}
\begin{remark}\label{88}
We may assume without loss of generality $V(0)<0$. So, by continuity of $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$, we can choose $r>0$ such that $V(x)\leq -\beta<0$ for $x \in B_{r}$ and some $\beta>0$.
\end{remark}

Consider the function
$$\varphi_{\varepsilon}(x):= \frac{c_{N}\psi(x)\varepsilon^{\frac{N-2}{2}}}{(\varepsilon^{2}+|x|^{2})^{\frac{N-2}{2}}}$$
where $c_{N}=(N(N-2))^{\frac{N-2}{4}}$, $\varepsilon>0$ and $ \psi \in C_{0}^{\infty}(\mathbb{R}^{N})$ is such that
$$\psi(x)=1\;\;\mbox{for}\;\; |x|\leq \frac{r}{2}\;\;\mbox{and}\;\; \psi(x)=0\;\;\mbox{for}\;\; |x| \geq r.$$
We shall need the following asymptotic estimates as $\varepsilon\rightarrow 0^{+}$ (see \cite{MW}).
\begin{eqnarray}
&& ||\nabla\varphi_{\varepsilon} ||_{2}^{2}=S^{\frac{N}{2}}+O(\varepsilon^{n-2}),\;\; ||\nabla\varphi_{\varepsilon} ||_{1}=O(\varepsilon^{\frac{N-2}{2}}),\;\; ||\varphi_{\varepsilon}||_{2^{*}}^{2^{*}}=S^{\frac{N}{2}}+O(\varepsilon^{N})\nonumber \\
&& \label{91} \\
&& ||\varphi_{\varepsilon}||_{2^{*}-1}^{2^{*}-1}= O(\varepsilon^{\frac{N-2}{2}}),\;\; ||\varphi_{\varepsilon}||_{1}=O(\varepsilon^{\frac{N-2}{2}})\nonumber
\end{eqnarray}
and
\begin{eqnarray} \label{94}
||\varphi_{\varepsilon}||_{2}^{2}=\left\{\begin{array}{c}
b \varepsilon^{2}|log(\varepsilon)|+O(\varepsilon^{2}),\;\;\mbox{if}\;\; N=4 \\
b \varepsilon^{2}+ O(\varepsilon^{N-2}),\;\; \mbox{if}\;\; N\geq 5.
\end{array}
\right.
\end{eqnarray}
where $b>0$.

\begin{proposition}\label{96}
	Suppose $N\geq 4$, there is $\varepsilon_{0}>0$ such that for all $\varepsilon \in (0, \varepsilon_{0})$ and for all  $O(\varepsilon^{N-2})$
	\begin{equation*}
	\frac{\displaystyle\int_{\mathbb{R}^{N}}(|\nabla\varphi_{\varepsilon}|^{2}+V(x)\varphi_{\varepsilon}^{2})}{|\varphi_{\varepsilon}|_{2^{*}}^{2}}+O(\varepsilon^{N-2})< S.
	\end{equation*}

\end{proposition}
\begin{proof}

	In fact, given $O(\varepsilon^{N-2})$
	\begin{eqnarray}
	\frac{\displaystyle\int_{\mathbb{R}^{N}}(|\nabla\varphi_{\varepsilon}|^{2}+V(x)\varphi_{\varepsilon}^{2})}{|\varphi_{\varepsilon}|_{2^{*}}^{2}}+O(\varepsilon^{N-2})&=& \frac{S^{\frac{N}{2}}+O(\varepsilon^{N-2})+\displaystyle\int_{\mathbb{R}^{N}} V(x)\varphi_{\varepsilon}^{2}}{[S^{\frac{N}{2}}+O(\varepsilon^{N})]^{\frac{N-2}{N}}}+O(\varepsilon^{N-2}) \nonumber \\
	&=& S\left[\frac{1+\frac{O(\varepsilon^{N-2})}{S^{\frac{N}{2}}}+\frac{1}{S^{\frac{N}{2}}}\displaystyle\int_{\mathbb{R}^{N}} V(x)\varphi_{\varepsilon}^{2}}{[1+\frac{O(\varepsilon^{N})}{S^{\frac{N}{2}}}]^{\frac{N-2}{N}}}\right]+O(\varepsilon^{N-2}) \nonumber \\
	&=&S\left[\frac{1+O(\varepsilon^{N-2})+\frac{1}{S^{\frac{N}{2}}}\displaystyle\int_{\mathbb{R}^{N}} V(x)\varphi_{\varepsilon}^{2}}{[1+O(\varepsilon^{N})]^{\frac{N-2}{N}}}\right]+O(\varepsilon^{N-2}). \nonumber
	\end{eqnarray}
	On the other hand, by (\ref{94}) and remark \ref{88}
	\begin{eqnarray} \label{95}
	\int_{\mathbb{R}^{N}} V(x) \varphi_{\varepsilon}^{2} \leq -\beta ||\varphi_{\varepsilon}||_{2}^{2}=\left\{\begin{array}{c}
	-\beta b \varepsilon^{2}|log(\varepsilon)|+O(\varepsilon^{2})\;\;\mbox{if}\;\; N=4 \\
	- \beta b \varepsilon^{2}+ O(\varepsilon^{N-2})\;\; \mbox{if}\;\; N\geq 5.
	\end{array}
	\right.
	\end{eqnarray}
	If $N=4$, by (\ref{95}), we obtain
	\begin{eqnarray}
	\frac{\displaystyle\int_{\mathbb{R}^{N}}(|\nabla\varphi_{\varepsilon}|^{2}+V(x)\varphi_{\varepsilon}^{2})}{|\varphi_{\varepsilon}|_{2^{*}}^{2}}+O(\varepsilon^{2}) &=& S\left[\frac{1+O(\varepsilon^{2})+\frac{1}{S^{2}}\displaystyle\int_{\mathbb{R}^{N}} V(x)\varphi_{\varepsilon}^{2}}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right]+O(\varepsilon^{2}) \nonumber \\
	&\leq & S\left[\frac{1+O(\varepsilon^{2})-\frac{1}{S^{2}}\beta b \varepsilon^{2}|log(\varepsilon)|+O(\varepsilon^{2})}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right]+O(\varepsilon^{2}) \nonumber \\
	&=& S\left(\frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right)+S\left(\frac{O(\varepsilon^{2})}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right) +\nonumber \\
	&-&S\left(\frac{\beta b \varepsilon^{2}}{S^{2}}\frac{|log(\varepsilon)|}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right) +O(\varepsilon^{2}). \nonumber
	\end{eqnarray}
	Note that
	\begin{itemize}
		\item [(1)]
		$\dfrac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\rightarrow 1$ as $\varepsilon \rightarrow 0^{+}$;
		\item [(2)] $\dfrac{O(\varepsilon^{2})}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}=O(\varepsilon^{2})$.
	\end{itemize}
		Just see that
		\begin{equation*}
		\frac{\frac{O(\varepsilon^{2})}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}}{\varepsilon^{2}}= \frac{O(\varepsilon^{2})}{\varepsilon^{2}} \cdot \frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}
		\end{equation*}
		is  bounded for $\varepsilon \approx 0^{+}$.

	Follow from $(1)$,  $\varepsilon \approx 0^{+}$, that
	\begin{equation*}
	\dfrac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}} \geq \frac{1}{2}\Leftrightarrow \dfrac{-\beta d \varepsilon^{2}}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}} \leq \frac{-\beta d \varepsilon^{2}}{2},
	\end{equation*}
	and with that
	\begin{eqnarray}
	\frac{\displaystyle\int_{\mathbb{R}^{N}}(|\nabla\varphi_{\varepsilon}|^{2}+V(x)\varphi_{\varepsilon}^{2})}{|\varphi_{\varepsilon}|_{2^{*}}^{2}}+O(\varepsilon^{2})&=& S\left(\frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right)+S\left(\frac{O(\varepsilon^{2})}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right)+\nonumber \\
	&-&S\left(\frac{\beta b \varepsilon^{2}}{S^{2}}\frac{|log(\varepsilon)|}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right)+O(\varepsilon^{2}) \nonumber \\
	&\leq & S\left(\frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right)+O(\varepsilon^{2})-S\left(\frac{\beta d \varepsilon^{2}}{2 S^{2}} |log(\varepsilon)|\right)+O(\varepsilon^{2}) \nonumber \\
	&= & S\left(\frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right)-S\left(\frac{\beta d \varepsilon^{2}}{2 S^{2}} |log(\varepsilon)|\right)+O(\varepsilon^{2}) \nonumber
	\end{eqnarray}
	Consider the application
	\begin{eqnarray}
	g: & [0, O(\varepsilon^{4})]& \longrightarrow \mathbb{R} \nonumber \\
	&t&\longmapsto g(t)= \frac{1}{[1+t]^{\frac{1}{2}}}. \nonumber
	\end{eqnarray}
	By the mean value theorem there is $\theta  \in (0, O(\varepsilon^{4}))$ such that
	\begin{eqnarray}
	\frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}- 1= -\frac{1}{2} [1+\theta]^{-\frac{3}{2}} O(\varepsilon^{4}), \nonumber
	\end{eqnarray}
	that is,
	\begin{equation*}
	\frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}= 1- \frac{1}{2} [1+\theta]^{-\frac{3}{2}} O(\varepsilon^{4})=1- O(\varepsilon^{4}).
	\end{equation*}
	Therefore,
	\begin{eqnarray}
	\frac{\displaystyle\int_{\mathbb{R}^{N}}(|\nabla\varphi_{\varepsilon}|^{2}+V(x)\varphi_{\varepsilon}^{2})}{|\varphi_{\varepsilon}|_{2^{*}}^{2}} +O(\varepsilon^{2})
	&\leq & S\left(\frac{1}{[1+O(\varepsilon^{4})]^{\frac{1}{2}}}\right)-S\left(\frac{\beta d \varepsilon^{2}}{2 S^{2}} |log(\varepsilon)|\right)+O(\varepsilon^{2}) \nonumber \\
	&=& S-O(\varepsilon^{4})-\frac{\beta d \varepsilon^{2}}{2 S} |log(\varepsilon)|+O(\varepsilon^{2}).\nonumber
	\end{eqnarray}
	\begin{claim}
	There is $\varepsilon_{0}>0$ such that
	$$O(\varepsilon^{4})+O(\varepsilon^{2})-\dfrac{\beta d \varepsilon^{2}}{2 S^{2}} |log(\varepsilon)|<0,\;\forall\;\varepsilon \in (0, \varepsilon_{0}).$$
	\end{claim}
	Follows from the fact that
	\begin{eqnarray}
	O(\varepsilon^{4})+O(\varepsilon^{2})-\dfrac{\beta d \varepsilon^{2}}{2 S^{2}} |log(\varepsilon)|&=& \varepsilon^{2} \left[\varepsilon^{2}\frac{O(\varepsilon^{4})}{\varepsilon^{4}}+\frac{O(\varepsilon^{2})}{\varepsilon^{2}}-\dfrac{\beta d}{2 S^{2}} |log(\varepsilon)| \right] \nonumber
	\end{eqnarray}
	with the fact that:
	\begin{equation*}
	\varepsilon^{2}\dfrac{O(\varepsilon^{4})}{\varepsilon^{4}}+\dfrac{O(\varepsilon^{2})}{\varepsilon^{2}} \;\; \mbox{is bounded for}\;\; \varepsilon\approx 0^{+}\;\;\mbox{and}\;\;\lim_{\varepsilon \rightarrow 0^{+}} -\dfrac{\beta d}{2 S^{2}} |log(\varepsilon)|= -\infty.
	\end{equation*}
	Therefore, there is $\varepsilon_{0}>0$ such that
	$$\frac{\displaystyle\int_{\mathbb{R}^{N}}(|\nabla\varphi_{\varepsilon}|^{2}+V(x)\varphi_{\varepsilon}^{2})}{|\varphi_{\varepsilon}|_{2^{*}}^{2}} +O(\varepsilon^{2})<S, \;\forall\; \varepsilon \in (0, \varepsilon_{0})\;\;\mbox{and for all}\;\; O(\varepsilon^{2}).$$
	The case $N \geq 5$ is analogous.
\end{proof}

\begin{remark}\label{92}

	\begin{itemize}

		\item [(1)] Since
		$$I_{\gamma}u)= \frac{1}{2} \int_{\mathbb{R}^{N}} (|\nabla u|^{2}+ V(x) u^{2})- \int_{\mathbb{R}^{N}} F_{\gamma}(x,u) \leq \frac{1}{2} \int_{\mathbb{R}^{N}} (|\nabla u|^{2}+ V(x) u^{2})- \frac{1}{2^{*}} \int_{\mathbb{R}^{N}} |u|^{2^{*}},$$
		defining
		$$J(u)=\frac{1}{2} \int_{\mathbb{R}^{N}} (|\nabla u|^{2}+ V(x) u^{2})- \frac{1}{2^{*}}\int_{\mathbb{R}^{N}} |u|^{2^{*}},$$
		we get $I_{\gamma}u) \leq J(u)$ for all $u \in H^{1}(\mathbb{R}^{N})$.
	\end{itemize}
\end{remark}
\begin{proposition}\label{97}
	For $u \in H^{1}(\mathbb{R}^{N})$, we have:
	\begin{itemize}
		\item [(i)] If $\displaystyle\int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)u^{2}] > 0$, then
		$$\max_{t\geq 0}J(tu)= \frac{1}{N} \left(\frac{\displaystyle\int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)u^{2}]}{||u||_{2^{*}}^{2}}\right)^{\frac{N}{2}}.$$
		\item [(ii)] If $\displaystyle\int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)u^{2}] \leq 0$, then
		$$\max_{t\geq 0}J(tu)=0.$$
	\end{itemize}
\end{proposition}
\begin{proof}

	Given $u \in H^{1}(\mathbb{R}^{N})$, set the function
	\begin{eqnarray}
	h: &[0, +\infty)&\longrightarrow \mathbb{R} \nonumber \\
	&t& \longmapsto h(t)=J(tu) \nonumber.
	\end{eqnarray}

	\textbf{Case (i)}.

	By the fact
	$$h(t)\rightarrow -\infty\;\; \mbox{as}\;\; t\rightarrow +\infty\;\; \mbox{and} \;\;h(t)>0 \;\; \mbox{for}\;\; t\approx 0,$$
	there is $t_{0} \in (0, +\infty)$ such that $h(t_{0})=\displaystyle\max_{t\geq 0} h(t)$. More precisely
	$$t_{0}= \left(\frac{\displaystyle\int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)u^{2}]}{||u||_{2^{*}}^{2^{*}}}\right)^{\frac{1}{2^{*}-2}}.$$
	Therefore,
	$$h(t_{0})=\frac{1}{N} \left(\frac{\displaystyle\int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)u^{2}]}{||u||_{2^{*}}^{2}}\right)^{\frac{N}{2}}.$$

	\textbf{Case (ii)}

	Just notice that $h(t) \leq 0$ for all $t\geq 0$ and $h(0)=0$.
\end{proof}
Before continuing we will make some considerations. First we wil need the following proposition, whose proof is in \cite[Proposition 2.2]{Chabrowski}.

\begin{proposition}\label{87}
	Suppose $V \in L^{\infty}(\mathbb{R}^{N})$ and $(V_{1})-(V_{2})$, there is $c_{0}>0$ such that
	$$||u^{-}||_{W^{1, \infty}(\mathbb{R}^{N})} \leq c_{0} ||u^{-}||_{2},\;\forall\; u^{-} \in E^{-}.$$
\end{proposition}

By the  convexity of the application $t\mapsto |t|^{2^{*}}$, H\"older inequality and Proposition \ref{87}, there is $c_{1}>0$ such that

\begin{eqnarray}
||u||^{2^{*}}_{2^{*}} &\geq& ||s\varphi_{\varepsilon}||_{2^{*}}^{2^{*}}+ 2^{*} \int_{\mathbb{R}^{N}} (s\varphi_{\varepsilon})^{2^{*}-1} u^{-} \nonumber \\
&\geq & ||s\varphi_{\varepsilon}||_{2^{*}}^{2^{*}}- c_{1}||\varphi_{\varepsilon}||^{2^{*}-1}_{2^{*}-1} ||u^{-}||_{2}\label{89}
\end{eqnarray}
and
\begin{equation}\label{90}
\int_{\mathbb{R}^{N}} (\nabla \varphi_{\varepsilon} \nabla u^{-} + V(x) \varphi_{\varepsilon} u^{-}) \leq O(\varepsilon^{\frac{N-2}{2}})||u^{-}||_{2}.
\end{equation}

\begin{proposition} \label{98}
	There is $\varepsilon_{0}>0$ such that
	$$\sup_{u \in Z_{\varepsilon}, ||u||_{2^{*}}=1} \int_{\mathbb{R}^{N}} (|\nabla u|^{2}+V(x) u^{2})<S, \;\forall\; \varepsilon \in (0, \varepsilon_{0}),$$
	where $Z_{\varepsilon}= E^{-} \oplus \mathbb{R} \varphi_{\varepsilon} \equiv E^{-} \oplus \mathbb{R} \varphi_{\varepsilon}^{+}$.
\end{proposition}
\begin{proof}

	Let $u= u^{-} + s \varphi_{\varepsilon}$ such that $||u^{-} + s \varphi_{\varepsilon}||_{2^{*}}=1$ . By (\ref{91}), (\ref{89}), (\ref{90}) and continuous Sobolev embeddings
	\begin{eqnarray}
	\int_{\mathbb{R}^{N}} (|\nabla u|^{2}+V(x) u^{2})&=& \int_{\mathbb{R}^{N}} (|\nabla (s\varphi_{\varepsilon})|^{2}+ V(x) (s\varphi_{\varepsilon})^{2})  + 2s\int_{\mathbb{R}^{N}} (\nabla \varphi_{\varepsilon} \nabla u^{-}+V(x) \varphi_{\varepsilon} u^{-}) + \nonumber \\
	&-& ||u^{-}||^{2}\nonumber \\
	&\leq & \frac{\displaystyle\int_{\mathbb{R}^{N}} (|\nabla \varphi_{\varepsilon}|^{2}+ V(x) \varphi_{\varepsilon}^{2})}{||\varphi_{\varepsilon}||_{2^{*}}^{2}} ||s\varphi_{\varepsilon}||_{2^{*}}^{2}+ 2s\;O(\varepsilon^{\frac{N-2}{2}})||u^{-}||_{2} - ||u^{-}||^{2} \nonumber \\
	&\leq & \frac{\displaystyle\int_{\mathbb{R}^{N}} (|\nabla \varphi_{\varepsilon}|^{2}+ V(x) \varphi_{\varepsilon}^{2})}{||\varphi_{\varepsilon}||_{2^{*}}^{2}} (1+ c_{1}||\varphi_{\varepsilon}||^{2^{*}}_{2^{*}} ||u^{-}||_{2}) +\nonumber \\
	&+&  2s\;O(\varepsilon^{\frac{N-2}{2}})||u^{-}||_{2} - \tilde{c}||u^{-}||_{2}^{2} \nonumber \\
	&\leq & \frac{\displaystyle\int_{\mathbb{R}^{N}} (|\nabla \varphi_{\varepsilon}|^{2}+ V(x) \varphi_{\varepsilon}^{2})}{||\varphi_{\varepsilon}||_{2^{*}}^{2}} + O(\varepsilon^{\frac{N-2}{2}})||u^{-}||_{2} -\tilde{c}||u^{-}||_{2}^{2} \nonumber
	\end{eqnarray}
	that is,
	$$\sup_{u \in Z_{\varepsilon}, ||u||_{2^{*}}=1} \int_{\mathbb{R}^{N}} (|\nabla u|^{2}+V(x) u^{2}) \leq \frac{\displaystyle\int_{\mathbb{R}^{N}} (|\nabla \varphi_{\varepsilon}|^{2}+ V(x) \varphi_{\varepsilon}^{2})}{||\varphi_{\varepsilon}||_{2^{*}}^{2}}+  O(\varepsilon^{\frac{N-2}{2}})||u^{-}||_{2}-\tilde{c}||u^{-}||_{2}^{2}.$$
	Knowing that
	$$ \alpha \cdot \beta \leq \frac{\alpha^{2}}{2}+ \frac{\beta^{2}}{2},\;\forall\; \alpha, \beta \geq 0,$$
	we get
	\begin{eqnarray}
	O(\varepsilon^{\frac{N-2}{2}})||u^{-}||_{2}&=& O(\varepsilon^{\frac{N-2}{2}})(\tilde{c})^{\frac{1}{2}} ||u^{-}||_{2} \nonumber \\
	&\leq& \frac{O(\varepsilon^{\frac{N-2}{2}})^{2}}{2}+ \frac{\tilde{c} ||u^{-}||_{2}^{2}}{2} \nonumber \\
	&=& \frac{O(\varepsilon^{N-2})}{2}+ \frac{\tilde{c} ||u^{-}||_{2}^{2}}{2}. \nonumber
	\end{eqnarray}
	\begin{eqnarray}
	\sup_{u \in Z_{\varepsilon}, ||u||_{2^{*}}=1} \int_{\mathbb{R}^{N}} (|\nabla u|^{2}+V(x) u^{2}) &\leq& \frac{\displaystyle\int_{\mathbb{R}^{N}} (|\nabla \varphi_{\varepsilon}|^{2}+ V(x) \varphi_{\varepsilon}^{2})}{||\varphi_{\varepsilon}||_{2^{*}}^{2}}+  \frac{O(\varepsilon^{N-2})}{2}+ \frac{\tilde{c} }{2}||u^{-}||_{2}^{2}-\tilde{c}||u^{-}||_{2}^{2} \nonumber \\
	&=& \frac{\displaystyle\int_{\mathbb{R}^{N}} (|\nabla \varphi_{\varepsilon}|^{2}+ V(x) \varphi_{\varepsilon}^{2})}{||\varphi_{\varepsilon}||_{2^{*}}^{2}}+  O(\varepsilon^{N-2})-\frac{\tilde{c} }{2}||u^{-}||_{2}^{2} \nonumber \\
	&\leq & \frac{\displaystyle\int_{\mathbb{R}^{N}} (|\nabla \varphi_{\varepsilon}|^{2}+ V(x) \varphi_{\varepsilon}^{2})}{||\varphi_{\varepsilon}||_{2^{*}}^{2}}+  O(\varepsilon^{N-2}). \nonumber
	\end{eqnarray}
	Therefore, by the Proposition \ref{96}, there is $\varepsilon_{0}>0$ such that
	$$\sup_{u \in Z_{\varepsilon}, ||u||_{2^{*}}=1} \int_{\mathbb{R}^{N}} (|\nabla u|^{2}+V(x) u^{2}) < S,\;\forall\; \varepsilon \in (0, \varepsilon_{0}).$$
\end{proof}

For $\varepsilon \in (0, \varepsilon_{0})$, by Proposition \ref{97} and \ref{98},  for $u \in Z_{\varepsilon}$,  we conclude
\begin{eqnarray}
J(u) &\leq& J(tu)\leq  \frac{1}{N} \left(\frac{\displaystyle\int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)u^{2}]}{||u||_{2^{*}}^{2}}\right)^{\frac{N}{2}} \nonumber\\
&\leq & \frac{1}{N} \left(\sup_{w \in Z_{\varepsilon}, ||w||_{2^{*}}=1}\displaystyle\int_{\mathbb{R}^{N}}[|\nabla w|^{2}+V(x)w^{2}]\right)^{\frac{N}{2}}, \nonumber
\end{eqnarray}
this is,
$$\sup_{u \in Z_{\varepsilon}} J(u) \leq \frac{1}{N} \left(\sup_{w \in Z_{\varepsilon}, ||w||_{2^{*}}=1}\displaystyle\int_{\mathbb{R}^{N}}[|\nabla w|^{2}+V(x)w^{2}]\right)^{\frac{N}{2}}< \frac{1}{N} S^\frac{N}{2}.$$
Since $c_{\gamma} \in [b_{\gamma}, d_{\gamma}]$ and
$$d_{\gamma}= \sup_{\mathcal{M}}I_{\gamma}$$
where
$$\mathcal{M}=\left\{u=u^{-}+t u^{+}\;; ||u||\leq \rho, t \geq 0 \;\;\mbox{and}\;\; u^{-} \in E^{-}\right\}$$
for some $u^{+} \in E^{+}\backslash \{0\}.$ We get $\mathcal{M} \subset Z_{\varepsilon}$ for $u^{+}= \varphi_{\varepsilon}^{+}$ and consequently
$$d_{\gamma} \leq \sup_{u \in Z_{\varepsilon}} J(u)<\frac{1}{N} S^\frac{N}{2}.$$
\begin{itemize}
	\item Case $N=3$.
\end{itemize}

\begin{remark}\label{111}
	Consider
	 $$
	F_{1}(x,t)= \left\{\begin{aligned}
	0,\;\;\mbox{if}\;\; |t|\leq a \\
	\frac{1}{q}|t|^{q}- \frac{1}{q}a^{q},\;\; \mbox{if}\;\; |t|>a.
	\end{aligned}
	\right.
	$$
Fixed $R>0$, for $x \in B_{R}$, we obtain:

		If $|u(x)|\leq a$, $F_{1}(x,u) = 0$ and
		$$\frac{1}{q}\int_{ B_{R}}|u(x)|^{q}\leq \frac{1}{q}a^{q}|B_{R}|.$$

		In case $|u(x)|>a$,
		$$\int_{ B_{R}}F_{1}(x,u)= \frac{1}{q}\int_{ B_{R}}|u(x)|^{q}-\frac{1}{q}a^{q}|B_{R}|.$$
		So,
		$$F_{1}(x,u)\geq \frac{1}{q}\int_{ B_{R}}|u(x)|^{q}-\frac{1}{q}a^{q}|B_{R}|.$$
\end{remark}

\begin{lemma}\label{106}
Given $z_{0} \in E^{+}\backslash \{0\}$ and $s_{0}>0$. Let $\rho>0$ given in the Lemma \ref{60}, there are $K>0$ and $R>0$ such that
	$$K||sz_{0}||_{L^{p}(B_{R})}\leq ||u^{-}+sz_{0}||_{L^{p}(B_{R})},$$
	$u^{-} \in E^{-}$, $s\geq s_{0}$, $p \in (2,2^{*})$ and $||u^{-}+sz_{0}|| \leq \rho$.
\end{lemma}
\begin{proof}

	Suppose that there are $s_{n} \geq s_{0}$, $u_{n}^{-} \in Y$ and $R_{n}\rightarrow +\infty$ such that
	$$ \left\| \frac{u_{n}^{-}}{s_{n}} +z_{0}\right\|_{L^{p}(B_{R_{n}})}=\frac{||u_{n}^{-}+s_{n}z_{0}||_{L^{p}(B_{R_{n}})}}{s_{n}}< \frac{||z_{0}||_{L^{p}(B_{R_{n}})}}{n},\;\forall\; n \in \mathbb{N}.$$
	So,
	\begin{equation}\label{109}
	\frac{u_{n}^{-}}{s_{n}}\longrightarrow -z_{0}\;\;\mbox{in}\;\;L^{p}(\mathbb{R}^{N}).
	\end{equation}
	On the other hand,
	$$\left\|\frac{u_{n}^{-}}{s_{n}}\right\|^{2} \leq \frac{||u_{n}^{-}+s_{n}z_{0}||^{2} }{s_{0}^{2}}\leq \frac{\rho^{2}}{s_{0}^{2}}.$$
	There is $w \in E^{-}$ such that, goingo to a subsequence if necessary,
	$$\frac{u_{n}^{-}}{s_{n}}\rightharpoonup w\;\;\mbox{in}\;\; H^{1}(\mathbb{R}^{N}),$$
	consequently
	\begin{equation}\label{110}
	\frac{u_{n}^{-}}{s_{n}}\longrightarrow w\;\; \mbox{in}\;\; L_{loc}^{p}(\mathbb{R}^{N}).
	\end{equation}
	By (\ref{109}) and (\ref{110}) we obtain
	$$w=-z_{0}\;\;\mbox{a.e in}\;\;\mathbb{R}^{N},$$
that is, $w=-z_{0} \in E^{+}\backslash \{0\}$ which contradicts the fact $w \in E^{-}$.

\end{proof}

\begin{lemma}\label{112}
	Given $z_{0} \in E^{+}\backslash\{0\}$, there is $s_{0}>0$ such that
	$$d_{\gamma}=\sup_{\mathcal{M}} I_{\gamma} = \sup_{A} I_{\gamma},$$
	where
	$$A=\{u^{-}+sz_{0}\;;\; ||u^{-}+sz_{0}||\leq \rho, u^{-} \in E^{-}\;\;\mbox{and}\;\; s\geq s_{0} \},$$
	and $\rho>0$ is given in the Lemma \ref{106}.
\end{lemma}
\begin{proof}

	By the definition of supreme $(s_{n}) \subset [0, +\infty)$ and $(u_{n}^{-})\subset E^{-}$ such that $||u_{n}^{-}+s_{n}z_{0}||\leq \rho$ and
	\begin{equation}\label{108}
	d_{\gamma}-\frac{1}{n} \leq I_{\gamma}(u_{n}^{-}+s_{n}z_{0})< \sup_{B_{\rho}\cap E} I_{\gamma}= d_{\gamma}>0\;,\;\forall\; n \in \mathbb{N}.
	\end{equation}
	\begin{claim}
	There is $s_{0}>0$ such that $s_{n}\geq s_{0}$ for all $n \in \mathbb{N}$.
	\end{claim}

	In fact, suppose that there is $(s_{n_{j}})\subset (s_{n})$ such that $s_{n_{j}}\rightarrow 0$, then
	$$I_{\gamma}(u_{n_{j}}^{-}+s_{n_{j}}z_{0})\leq \frac{s_{n_{j}}^{2}}{2}||z_{0}||^{2}\rightarrow 0.$$
	Choosing $n_{j_{0}} \in \mathbb{N}$ such that $\frac{s_{n_{j}}^{2}}{2}||z_{0}||^{2}<\frac{d_{\gamma}}{2}$, for $n_{j} \geq n_{j_0}$, we obtain
	$$d_{\gamma}-\frac{1}{n_{j}} < \frac{d_{\gamma}}{2}$$
	what contradicts (\ref{108}).

	So $u_{n}^{-}+s_{n}z_{0} \in A$ and
	$$d_{\gamma} \geq \sup_{A}I_{\gamma} \geq I_{\gamma} (u_{n}^{-}+s_{n}z_{0})=\sup_{B_{\rho}\cap E}I_{\gamma}+o_{n}(1)= d_{\gamma}+o_{n}(1).$$
\end{proof}

\begin{lemma} There is $\gamma>0$ such that
	$$\sup_{u \in A}I_{\gamma}(u)< \dfrac{S^{\frac{N}{2}}}{N },$$
	where $A$ is given in the Lemma \ref{112}. In addition, for this is $\gamma>0$,
	$$c_{\gamma} \leq d_{\gamma}< \dfrac{S^{\frac{N}{2}}}{N }.$$
\end{lemma}

\begin{proof}

	Since $I_{\gamma}(u) \leq J_{\gamma}(u)$ for all $u \in H^{1}(\mathbb{R}^{N})$, where
	$$J_{\gamma}(u)= \frac{1}{2}\int_{\mathbb{R}^{N}} (|\nabla u|^{2}+V(x)u^{2})- \gamma \int_{\mathbb{R}^{N}}F_{1}(x,u),$$
	so just prove the estimate to $J_{\gamma}$.

	Let $u \in A$, by Remark \ref{111} and Lemma \ref{106}, there are $K>0$ and $R>0$ such that
	\begin{eqnarray}
	J_{\gamma}(u)&\leq & \frac{s^{2}}{2}||z_{0}||^{2}-\frac{1}{2}||u^{-}||^{2}- \frac{\gamma}{q} \int_{B_{R}} |u^{-}+sz_{0}|^{q}+\frac{\gamma}{q}a^{q}|B_{R}| \nonumber \\
	&\leq & \frac{s^{2}}{2}||z_{0}||^{2}- \frac{K\gamma}{q} \int_{B_{R}} |sz_{0}|^{q}+\frac{\gamma}{q}a^{q}|B_{R}|, \nonumber
	\end{eqnarray}
	that is,
	$$\sup_{u \in A}I_{\gamma}(u) \leq \sup_{s\geq s_{0}} \left(\frac{s^{2}}{2}||z_{0}||^{2}- \frac{K\gamma s^{q}}{q} \int_{B_{R}} |z_{0}|^{q}+\frac{\gamma}{q}a^{q}|B_{R}|\right).$$
	Set the function
	\begin{eqnarray}
	h:&[0, +\infty)&\longrightarrow \mathbb{R} \nonumber \\
	&s& \longmapsto h(s)=\frac{s^{2}}{2}||z_{0}||^{2}- \frac{K\gamma s^{q}}{q} ||z_{0}||_{L^{q}(B_{R})}^{q} +\frac{\gamma}{q}a^{q}|B_{R}|\nonumber.
	\end{eqnarray}
	Note that
	\begin{itemize}
		\item $h(s)\rightarrow -\infty$ as $s\rightarrow +\infty$;
		\item $h(s)>0$ for $s\approx 0^{+}$.
	\end{itemize}
	Therefore, there is $t_{0} \in (0, +\infty)$ such that $h'(t_{0})=0$, this is,
	$$t_{0}= \left(\frac{||z_{0}||^{2}}{K\gamma ||z_{0}||_{q}^{q}}\right)^{\frac{1}{q-2}}.$$
	So,
	$$\max_{s\geq 0}h(s)=h(t_{0})=\left(\frac{1}{2}-\frac{1}{q}\right) \left(\frac{||z_{0}||^{2}}{||z_{0}||_{L^{q}(B_{R})}^{2}}\right)^{\frac{q}{q-2}} \left(\frac{1}{K\gamma}\right)^{\frac{2}{q-2}}+ \frac{\gamma}{q}a^{q}|B_{R}|.$$
	Fixed
	$$\gamma > \left(\frac{1}{2}-\frac{1}{q}\right)^{\frac{q-2}{2}} \left(\frac{||z_{0}||^{2}}{||z_{0}||_{L^{q}(B_{R})}^{2}}\right)^{\frac{q}{2}} \left(\frac{1}{K}\right)^{\frac{2}{q-2}} \left(\frac{2N}{S^{\frac{N}{2}}}\right)^{\frac{q-2}{2}},$$
	we obtain
	$$\max_{s\geq 0}h(s)=h(t_{0})< \frac{S^{\frac{N}{2}}}{2N}+ \frac{\gamma}{q}a^{q}|B_{R}|.$$
	Lastly, choosing $a \geq 0$ such that
	$$0\leq a \leq  \left(\frac{q S^{\frac{N}{2}}}{2 \gamma N |B_{R}|}\right)^{\frac{1}{q}}$$
	we conclude that
	$$\max_{s\geq 0}h(s)=h(t_{0})< \frac{S^{\frac{N}{2}}}{2N}+\frac{S^{\frac{N}{2}}}{2N}= \frac{S^{\frac{N}{2}}}{N} .$$

\end{proof}

	Now our goal is to prove that
	$$-\Delta v(x) + V(x) v(x) \in \partial_{t} F_{\gamma}(x,v(x))\;\;\mbox{a.e in}\;\; \mathbb{R}^{N},$$
	where $v$ is the weak limit of $(v_n)$ in $H^{1}(\mathbb{R}^N)$.

	From the study above, there exists $(\tilde{\omega}_{n}) \subset \partial I_{\gamma}(v_{n})$ such that $\tilde{\omega}_{n}= Q'(v_{n})- \tilde{\rho}_{n}$ and $||\tilde{\omega}_{n}||_{*}=o_{n}(1)$ where $(\tilde{\rho}_{n}) \subset \partial \Psi_{\gamma}(v_{n})$. For $\phi \in H^{1}(\mathbb{R}^{N})$, we obtain
	\begin{eqnarray}
	\left <\tilde{\rho}_{n}, \phi \right>= \left <Q'(v_{n}), \phi \right>-\left <\tilde{\omega}_{n}, \phi \right>\rightarrow \left <Q'(v), \phi \right>, \;\mbox{as}\;\;n\rightarrow +\infty,\nonumber
	\end{eqnarray}
	that is, $\tilde{\rho}_{n} \stackrel{*}{\rightharpoonup} Q'(v)$ in $(H^{1}(\mathbb{R}^{N}))^{*}$. Then, by Proposition \ref{7},  $Q'(v) \in \partial \Psi_{\gamma} (v)$. Thereby, $Q'(v)= \rho \in \partial \Psi_{\gamma} (v)$, and so,
	$$\int_{\mathbb{R}^{N}}(\nabla v \nabla \phi + V v \phi)= \int_{\mathbb{R}^{N}} \rho \phi\;\;\mbox{for all}\;\;\phi \in H^{1}(\mathbb{R}^{N}),$$
	where $\rho(x) \in \partial_{t} F_{\gamma}(x,v(x))\;\;\mbox{a.e in}\;\; \mathbb{R}^{N}.$ Hence
	$$
	\left\{\begin{aligned}
	-\Delta v + V(x)v &= \rho(x)\;\;\mbox{in}\;\;\mathbb{R}^{N}, \\
	v \in H^{1}(\mathbb{R}^{N}).
	\end{aligned}
	\right.
	$$
	Since $\rho \in L_{loc}^{\frac{2N}{N+2}}(\mathbb{R}^{N})$, the elliptic regularity theory gives that $v \in W_{loc}^{2,\frac{2N}{N+2}}(\mathbb{R}^{N})$ and
	$$-\Delta v + V(x)v = \rho(x)\;\;\mbox{a.e in}\;\; \mathbb{R}^{N},$$
	that is,
	$$
	-\Delta v(x) + V(x) v(x) \in \partial_{t} F_{\gamma}(x,v(x))\;\;\mbox{a.e in}\;\; \mathbb{R}^{N},
	$$
	finishing the proof of Theorem \ref{Teorema1}.

redpajama_set_name: RedPajamaArXiv


Ben and Kate discover an unopened envelope from 15 years ago that contains the yearly scavenger hunt their dad used to organize for them when they were kids. The find prompts them to complete the unfinished hunt, but raises old issues of sibling rivalry. Meanwhile, the scavenger hunt takes an unexpected detour for Kate when BJ enlists her to help get belongings back from an old fling.


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text:
Reports | Hasan Ali to return for Multan Test against England in place of injured Haris Rauf
>sports
Hasan Ali is likely to be added to Pakistan squad for England series
(PCB)
As per a report published by ESPNcricinfo, Hasan Ali is likely to be added to the squad replacing Haris Rauf for the second Test in the England series. Rauf sustained an injury while fielding during Rawalpindi Test which culminated in a grade two strain in the pacer's right squad.
Pakistan's performance in the first Test of a three-match series against England has come under severe scrutiny as they suffered a 74-run defeat on a flat deck. The bowling unit lacked experience as three bowlers were making debuts in the game. The hosts' problems further escalated as Haris Rauf picked up an injury during the first Test leading to squad strain and was ruled out of the series. However, the team has come up with a solution to the problem according to a report by ESPNcricinfo as they want to add some experience to their bowling attack and Hasan Ali is likely to replace Rauf for the remaining two Tests.
"Pakistan are likely to call up fast bowler Hasan Ali to the Test squad to replace Haris Rauf, who has been ruled out of the rest of the England series with a quad injury. Rauf was injured while fielding during England's first innings in Rawalpindi, leading to a grade two strain in his right quad," the report read.
Hasan was a mainstay on the national side till 2021 when he was Pakistan's Test Cricketer of the Year. However, his Test form dipped this year and he hasn't been a regular presence on the national side. Hasan scalped 14 wickets in the recently concluded Quaid-e-Azam trophy but it took 40 runs for each wicket to fall.
Mohammad Abbas is another option for the team as he had an impressive first-class season recently taking 18 wickets at 24.16 for Sindh. However, the report reveals that he hasn't been contacted yet and so Hasan Ali is likely to be added to the squad.
Haris Rauf
Hasan Ali
Pakistan Cricket Team
England Cricket Team

redpajama_set_name: RedPajamaCommonCrawl


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Structure - Leather isn't made up of a network of fibres like fabrics are. It has a dense structure. This has implications for how it is used. The most important thing to note when working with leather is that unlike most fabrics, piercing leather with a needle leaves a permanent and visible mark in the material. On a more positive note, it also means no frayed ends!
Size - Leather can be quite thick - it typically has a thickness between 1mm and 3mm. A full leather hide is the largest single piece of leather you can get (limited by the size of the animal it came from!). Check out this resource for more information on leather sizes and weights.
Natural Product - Real leather is a natural product made from animal hides, and as such will contain imperfections and blemishes. Although these can add charm to your project, you will probably want to hide unsightly blemishes. This means you will need to plan ahead when cutting your material.
Friction - Leather has a sticky feel to it which can make it difficult to feed evenly through your sewing machine.
Cost - Leather is very expensive. A full hide could cost upwards of £100. This means that you need to plan well before you start sewing.
Faux leather can be a good substitute for real leather in some cases. It is generally cheaper and easier to clean than real leather, and is blemish-free. But it is also likely to be less durable, and you simply can't beat a product made from real leather. The amazing smell of real leather is another selling point!
It is well known that most household sewing machines aren't powerful enough to take on leather. You will be left with many broken needles and skipped stitches, and you might even damage your machine in the process.
Industrial sewing machines are without question the best option when it comes to leather. They provide the required power and stability to get perfect stitches every time. But we all know that industrial machines are expensive, and it tends to be impractical to have one in your home.
This is why heavy duty domestic sewing machines are the perfect compromise. They are powerful enough to take on thin and medium weight leathers, they don't take up too much space, and they remain very affordable.
Powerful Motor - A powerful motor is essential to get through multiple layers of leather. Most domestic computerised machines these days are lightweight and do not have very powerful motors, so I would always recommend a mechanical machine for working with leather. Mechanical machines tend to pack that extra punch.
Sturdy Machine - Avoid lightweight plastic machines. These will jump about and you may end up ruining a lot of leather. Look for heavy machines which have metal frames. These are normally very stable and durable.
Leather Needles - Using needles suitable for leather work is crucial for getting good results. More about this later.
Suitable Presser Feet - Making sure you are using the most suitable presser feet will really help you deal with the stickiness of leather. I have a separate section on presser feet further down, as a wise choice of presser foot will make the world of difference.
Suitable Thread - Lightweight thread won't do the job. I normally recommend using good quality Nylon thread which is relatively thick when working with leather. Definitely experiment using scraps and different types of thread before starting your project in earnest.
Adjustable Presser Foot Pressure - Being able to adjust the presser foot pressure can help you get a perfect feed through the machine.
High Presser Foot Lift - Having a high presser foot lift will allow you to get a few extra layers under the foot.
I've said it before but it really can't be repeated too often - making sure you're using the right needles is step 1 when getting set up for a project. This is particularly true when working with leather!
The reason for this is quite simple - when a needle passes through woven and knitted fabrics it doesn't normally cut directly through the fabric. Rather it pushes the strands or yarns of the fabric to either side to find a way through. Needles with a spear point or ball point are suitable for this.
Leather isn't made up of individual strands, and so the needle will need to slice directly through the material. The cutting point on leather needles is designed specifically for this purpose. This type of needle point is called wedge point.
Your standard needle will break in no time when up against a thick piece of leather, but a leather needle will slice through without trouble. Keep an eye on your needle as you go though - you may need to change it more frequently than you are used to.
You will also need to think about the size of your leather needle. Usually a size 90/14 or 100/16 will be suitable. But remember that any hole the needle makes is permanent. For this reason you want to make sure you don't use a needle which is too large.
For more detail about choosing a needle, have a read of my guide to sewing machine needles.
Teflon/Silicone/Non-Stick Presser Foot - A non-stick foot has a slippery surface which will help feed the sticky leather through. Best of all, they are incredibly cheap (click here to buy one on Amazon.co.uk).
Roller Presser Foot - The roller presser foot works in a similar way to the non-stick foot. It has built-in textured rollers which will help feed the leather through the machine. This foot is always a great option when working with fabrics that are difficult to feed.
Walking Foot - If you have a lot of layers and need that extra bit of help getting the leather fed through, a walking foot is essential. It will help drag the layers of leather through the machine seamlessly.
Quite a few people have told me that in a bind you can stick a piece of Sellotape to the bottom of your all-purpose foot to avoid the leather sticking. I haven't tried it myself, but I like the idea! Let me know in the comments if it works.
If you are new to sewing and are a bit lost, I talk more about different types of presser feet in my beginner's guide to sewing machines.
If you've never worked with leather before one of the best ways to get used to working with it is to start by following a simple tutorial. There are loads of them out there on the Internet, but my favourites include this clutch bag, this beautiful leather journal cover, and this iPhone cover.
Once you've got the hang of it, how about trying a leather jacket!
Those of you who are interested in leather upholstery should take a look at my guide to sewing machines for upholstery.
I have selected 4 different heavy duty sewing machines which are perfect for working with leather. All of these machines are available for purchase in the UK.
The Janome 423S is a powerful and very robust machine. It has a metal frame which won't warp like plastic-bodied machines, so you can be sure it will remain stable when up against thick materials.
It offers a total of 24 stitches, including a 1-step buttonhole, and has adjustable presser foot pressure, which will help fine tune feeding leather through the machine.
Other benefits of this machine include the fact that it comes with a hard cover, and has a built-in needle threader.
The Janome HD2200 is similar to the 423S in many ways. It also has a sturdy metal frame, adjustable presser foot pressure, and a needle threader. It has a total of 22 stitches.
The HD2200 has the upper hand in terms of power though. It packs more punch that the 423S. It is slightly more expensive though.
The Singer Heavy Duty 4411 is a great machine which is certainly powerful.
It only has 11 stitches, which some may consider a drawback compared to the other machines. Bear in mind though that when working with leather you will mostly be using straight stitches, and are unlikely to need decorative stitches at all.
The Singer Heavy Duty 4423 machine is very similar to the 4411 model. However it has a total of 23 stitches, adding 12 that the 4411 doesn't have, which are mostly stretch and decorative stitches. It also has an automatic needle threader, which is missing from the 4411 model.
Measure everything twice! Cutting something to the wrong size could be an expensive mistake!
Pay close attention to thread tension. If your stitches don't look right, thread tension is likely to be the culprit. Perfect your thread tension by testing on scraps first.
Use longer stitches with leather. If your stitch length is short you will be perforating the leather too frequently, which could weaken it.
If you just want to practice working with leather or are working on a small project there's no need to go out and buy a full leather hide. You can buy scrap bundles relatively cheaply online. It pays to shop around too!
Depending on your project, hammering the seams with a rubber mallet can set them into the leather and give a nice finish.
You cannot use pins to hold your pieces of leather together whilst sewing, as they would create permanent holes. Instead you can use double sided tape, clips or leather glue. Using clips is my preferred option as it is the least messy. You can even use paperclips in a bind.
Use a leather marking pen or chalk to mark your leather, as this won't leave a mark.
If you're not used to working with leather, take it slow in the beginning to avoid making mistakes!
Well that about sums it up. Let me know in the comments if you have any questions or anything to add!


You can buy a house with much less cash than you think.
Local or national programs (such as the First Time Home Buyer's Program) exist to help access the real estate market. You can also qualify as a first time home buyer even if your spouse owned a house before, as long as your name was not registered as co-owner. Make sure your agent is well-informed and knowledgeable about the home buyer's programs in order to offer you all the possibilities.
You could get some help from your financial institution for your initial deposit and acquisition costs.
Even if you don't have the initial deposit available, if you don't have debts and some net worth (like a fully paid car), your financial institution might lend you the funds for your initial deposit which would be secured by that asset.
Some sellers might grant a second legal hypothec lien. In this case, the seller becomes more or less the lending institution. Instead of paying him⁄her the cash for the house, you pay monthly payments.
By borrowing money to invest in an RRSP until the required amount is achieved, you can benefit from a tax credit that you will use as cash. It is true that the money borrowed can be technically considered as a personal loan, but the monthly payment might be lower. Then, the money invested in the house and in the RRSP is yours.
If you can't get the minimum amount in cash or provide security for a loan because your net worth is too low, lending institutions will still accept your mortgage request.
This is easy to do and will provide you with peace of mind when the time comes to buy a house. Mortgage brokers can help you obtain an approval in writing at no cost and with no obligation. This can even be done over the phone. Better than a verbal approval, written pre-qualification is like having cash in the bank account. You will receive a certificate that guarantees the amount of your mortgage loan; very useful when you finally find the house you were looking for.Consider asking a professional specialized in mortgage loans. Using his or her services can make the difference between obtaining a mortgage and remaining a tenant forever. Usually, there are no fees to obtain the information. Then why on earth would you continue to waste thousands of dollars in rent when you could take a few minutes with your agent to talk about your specific needs in order to stop renting an own a house.
This conversation will not cost you anything. And of course, you should not feel obliged to buy a house at the moment you read these lines, but take the time to explore various opportunities, discover ways to own a house, imagine how informed and relaxed you will feel when the time comes to make this important decision.
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Summer's wrapping up, a new academic year's about to roll in, and if all goes well, I might be returning to more active blogging around here.
I have also returned, about a week ago, from the 12th International Congress for Finno-Ugric Studies. You can check out my presentation online, too: Semivowel losses and assimilations, in Finnic and beyond. Longtime readers may recall me having first explored the ideas within many blog posts (and one blog platform) ago. Evidence has continued to turn up, and I'm by now quite convinced that my newfound soundlaw *je- > Finnic *i- indeed exists.
The title is admittedly a bit more general than what might be warranted per the presentation's contents. For space concerns, I was not able to treat the topic of initial semivowels in Uralic languages in more general.
I could mention one fairly simple addendum here, though — while *wo- has been already traditionally well-established, and I attempt to show that abundant evidence of *je- can be found as well, by contrast it seems to me that **wu- and **ji- were not possible sequences in Proto-Uralic (and they remain impossible in most descendant languages as well).
Only one widely accepted instance *wu- has been proposed: the word for 'new' (> Fi. uusi, Hu. új etc.), traditionally reconstructed (modulo notation) as *wud₂ə. I however suspect that this should be instead reconstructed with *o, and that the evidence suggesting *u is due to the shift *o-ə > *u-ə in open syllables, as first proposed by Janhunen (1981) for the Finno-Permic end of the family. (Seen also in e.g. *lomə > *lumə > Fi. lumi 'snow'. Sammallahti has later suggested that the change also affected Ugric; I am skeptical, however. More on this at some point in the future.) — For *ji- there are a few more potential examples, but the best-looking cases (? *jikä 'age', ? *jitɜ 'night') fall among those where I believe *je- should be rather reconstructed.
If so, then it seems to me that we can likely apply for Proto-Uralic a phonological analysis also known from various other languages of the world: to unite *j *w on one hand and *i *u on the other as allophones of each other.
Nice! I quite like the idea that PIE *ǵ was loaned as PU *j.
On the slide about ien, you call *kn > *ŋ "an extraordinary sound shift". Is it extraordinary at the ends of words? In southeastern German dialects like mine, -gen /gn̩/ regularly becomes [ŋ̩] (legen [ˈlẽŋ̩]); the resulting sequence of a vowel and a syllabic consonant loses the latter's syllabicity in more common words, like sehen > Verner variant *segen > [sɛ̃ŋ]. Importantly, /g/ is voiceless, so it's the same sound as generic Uralic /k/ (or Spanish /k/ for that matter).
Likewise -ben, -den > [m̩], [n̩].
It would be extraordinary in the overall picture of the consonant history of Permic; normally all medial single stops are simply lost, plus there does not seem to have been any syncope that would have produced that type of awkward consonant clusters in the first place.
Are there any examples of PIE *g being borrowed into PU?
The only comparison involving PIE *g (which was, after all, kind of on the rare side) that I can find offhand is PIE *mozg- ~ PU *mośkə- 'to wash', but the palatalized *ś makes this difficult to analyze as a loan.
I see. Are there any examples of *k or *gʰ (both also on the rare side)?


Our doorstops are made from beautifully designed quality fabrics.
Please note: The doorstop arrives unfilled to reduce excessive postage costs.
A zip at the side makes it easy to fill with your choice of dried wheat, rice, sand (enclosed in a plastic bag), pebbles or stones plus some toy filling, cushion filling or fabric scraps.
Limited stock in some designs!


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The History of the Grey CupNovember 25, 2013
The History of the Grey Cup
The first Grey Cup game was held on December 4, 1909 when the University of Toronto Varsity Blues defeated the Parkdale Canoe Club 26–6 before 3,800 fans. They retained the trophy in the following two years, defeating the Hamilton Tigers in 1910 and the Toronto Argonauts in 1911.
Canada's participation in the First World War resulted in the cancellation of the championship from 1916 to 1918, during which time the Cup was forgotten. Montreal Gazette writer Bob Dunn claimed that the trophy was later rediscovered as "one of the family heirlooms" of an employee of the Toronto trust company where it had been sent for storage. The Grey Cup game was also cancelled in 1919 and finally resumed in 1920 with the 8th Grey Cup game, won 16–3 by the Varsity Blues over the Argonauts. It was the University of Toronto's fourth, and final, championship.
Operations were again suspended in 1942 due to the Second World War. Grey Cup play was expected to be suspended along with the unions however the military felt the game and sport would serve as a morale booster and organized teams at bases across the country.
As the 1950s wore on, the Canadian Football Council was formed in 1956 to administer the game at the professional level. Two years later, on January 18, 1958, the Canadian Football Council (CFC) withdrew from the Canadian Rugby Union (CRU) and reorganized as the Canadian Football League (CFL). The new league formally assumed control of the Grey Cup.
The Toronto Argonauts have won the most Grey Cup championships. The Saskatchewan Roughriders have the most losses in Grey Cup play (15), including five consecutive losses between 1928 and 1932.
The Grey Cup is the name of both the championship game of the Canadian Football League (CFL) and the trophy awarded to the victorious team. It is contested between the winners of the CFL's East and West Divisional playoffs and is one of Canadian television's largest annual sporting events. The 101st Grey Cup was played on November 24, 2013 in Regina, Saskatchewan where the Saskatchewan Roughriders defeated the Hamilton Tiger-Cats (45 - 23).
Effort Trust would like to congratulate the Roughriders on their big win and hopefully, we will see Hamilton in the Grey Cup next year and see them bring home the cup.
Source - Wikipedia
Effort Trust, Grey Cup


In first time , could you remove the folder , with our MOD MW .
I supose you enter conflict , with an old MOD.
I have removed all the folders of MW2 from the profile of brut45uk and still it will not work.
If I connect under another profile "exchalibur" there is no problem and I join your server.
Below is an image of what I receive.


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Statement From State's Attorney Berlin Regarding Naperville Police Officer-Involved Shooting of Male Armed with Hatchet
Mon Aug 15, 2022 09:34 pm Cardinal News CRIME BLOG C, Law Comments Off on Statement From State's Attorney Berlin Regarding Naperville Police Officer-Involved Shooting of Male Armed with Hatchet
"Every case involving the use of deadly force by a police officer, whether on or off duty, must be carefully and thoroughly investigated. Such scrutiny is required to ensure the protection of the civil rights of those involved and to maintain the public's confidence in law enforcement.
After a thorough and extensive investigation surrounding the shooting of twenty-eight-year-old Edward Samaan, which occurred on June 3, 2022, in the late morning hours at the intersection of Bond and McDowell Street, Naperville, by an on-duty Naperville Police Officer, it is my determination that the officer's actions were justified and no criminal charges will be filed against the Officer.
Body-worn video (YouTube.com PoliceActivity BodyCam & DashCam 14:57 combined elapsed)
In reaching this conclusion, my staff and I carefully reviewed the applicable law and thoroughly examined all of the evidence, including but not limited to:
Police Reports;
Statements from those involved;
Body-worn video;
In-car squad video;
Witness accounts;
Site visits;
Physical evidence;
Analysis of physical evidence.
On Friday, June 3, 2022, at approximately 11:00 a.m., Naperville Police Officer Frank Tonkovich, observed a black Honda Civic disobey a stop sign at eastbound McDowell Road and Bond Street. Officer Tonkovich, who was driving a marked Naperville Police squad car, positioned his vehicle behind the Honda and conducted a traffic stop on the Honda Civic on northbound Bond St. just north of McDowell Road.
At approximately 11:13 a.m., while Officer Tonkovich was speaking with the driver of the Honda Civic, a silver Ford Fusion driven by Edward Samaan stopped in the middle of the road within just a few feet of Officer Tonkovich's location. Samaan exited the Ford Fusion with a hatchet in his right hand and charged at Officer Tonkovich. Officer Tonkovich drew his firearm and fired six shots at Samaan, killing him. Samaan was shot 5 times. Samaan died from the gunshot wounds. An investigation into the shooting was conducted by the MERIT Public Integrity Team. MERIT investigators processed the scene where they recovered six spent shell casings. MERIT investigators also recovered the hatchet carried by Samaan.
The above facts have been evaluated in the context of Illinois law governing the justifiable use of deadly force. In accordance with Illinois law, my staff and I have reviewed the facts and circumstances of the case with special consideration given to the perspective of the officer on the scene. It is important to remember that police officers are often forced to make split-second decisions about the appropriate amount of force necessary to bring a tense, uncertain and rapidly evolving situation under control.
When Officer Tonkovich was speaking with the driver of the Honda Civic, a silver Ford Fusion driven by Edward Samaan stopped in the middle of the road within just a few feet of Officer Tonkovich's location. Samaan exited the Ford Fusion with a hatchet in his right hand and charged at Officer Tonkovich. Officer Tonkovich noted that "prior to the deadly force situation, when Edward Samaan abruptly stopped his vehicle next to me, he had a crazed look in his eyes. He exited from his vehicle in an aggressive manner with a hatchet and immediately and violently charged at me." Officer Tonkovich wrote in his report that he "fear[ed] for my life" and the "life of (the driver of the Honda Civic)." Officer Tonkovich wrote in his report, "I was forced to draw my firearm and use deadly force to stop Samaan from killing me or (the driver of the Honda Civic)."
According to the U.S. Supreme Court, the "reasonableness of a particular use of force must be judged from the perspective of a reasonable officer on the scene, rather than with the 20/20 vision of hindsight. Graham v. Connor, 490 U.S. 389 at 396 (1989). "[H]owever, the 'reasonableness' inquiry in an excessive-force case is an objective one: the question is whether the officers' actions are 'objectively reasonable' in light of the facts and circumstances confronting them, without regard to their underlying intent or motivation." Graham, 490 U.S. at 397. Under the new version of Illinois Statute 720 ILCS 5/7-5(f), Peace officer's use of force in making arrest, "the decision by a peace officer to use force shall be evaluated from the perspective of a reasonable officer in the same situation, based on the totality of the circumstances known to or perceived by the officer at the time of the decision, rather than with the benefit of hindsight, and that the totality of the circumstances shall account for occasions when officers may be forced to make quick judgments about using force."
When Edward Samaan drove up to Officer Tonkovich during a traffic stop of an unrelated person, exited from his vehicle in an aggressive manner with a hatchet and immediately and violently charged at Officer Tonkovich, Officer Tonkovich reasonably believed that Edward Samaan was trying to kill him or the subject of the traffic stop, and reasonably believed deadly force was necessary to prevent imminent death or great bodily harm to himself or the driver of the vehicle he had stopped. Moreover, evaluating Officer Tonkovich's actions from the perspective of a reasonable officer in the same situation, based on the totality of the circumstances known to or perceived by the officer at the time of the decision, rather than with the benefit of hindsight, Officer Tonkovich's action of discharging his duty weapon at a man charging at him and the motorist he had lawfully stopped while armed with a hatchet, a deadly weapon, was not only reasonable, but necessary in order to prevent great bodily harm or death to himself or the motorist he had stopped.
It is indeed a tragedy that previous hospitalizations and treatment efforts were unsuccessful in preventing Edward Samaan from engaging in violent behavior. Given the actions of Edward Samaan on the early morning of June 3, 2022, and Samaan's act of charging towards Officer Tonkovich with a hatchet in his hand, it was reasonable for Officer Tonkovich to believe Edward Samaan was trying to kill him and/or the motorist he had lawfully stopped, and that deadly force was necessary to prevent imminent death or great bodily harm to himself or others. Therefore, based on the results of an incredibly thorough investigation by the MERIT Public Integrity Team, it is the conclusion of my Office that Officer Tonkovich acted lawfully under Illinois Statutes and was justified in using deadly force when he fired his service weapon and shot Edward Samaan.
Officer Tonkovich and all of the responding Naperville Police Officers should be commended for their professionalism during this extremely tense incident. Their concern for the safety and well-being of Edward Samaan after he had been shot is a testament to the excellent training they received from the Naperville Police Department. Our sympathies, condolences and thoughts go out to the family of Edward Samaan as they grieve the loss of a loved one."
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DuPage County State's Attorney Robert B. Berlin
DuPage County State's Attorney's Office
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MERIT Public Integrity Team
police-involved shooting
police-involved shooting Naperville
Arrests in Dolton Include Wanted Chicagoan Accused of Aggravated Vehicular Hijacking in New Lenox
Fatal Wrong-Way Crash on Route 41 Near Wadsworth Rd Wadsworth


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Benedict News Online
Editorial: Publicize St. Benedict's Prep's Financial Aid
St. Benedict's Prep must advertise its generous financial aid packages to avoid turning away parents in light of its significant tuition increase
The Benedict News urges the Admissions Office and administrators to make a concerted effort to show that financial circumstances should not prevent a boy from receiving a valued education at St. Benedict's Prep. That generous financial aid has always been offered here and will continue to be is a crucial message as the tuition increases.
As reported by Mahishan Gnanaseharan, the change will increase the tuition to $12,500. Robert DiQuollo, the Vice Chairman of Finance Committee of the Board of Trustees, said that the change was appropriate in order to "accurately reflect the value of an education at St. Benedict's Prep." Mr. DiQuollo, along with faculty and other board members on a tuition committee, compared the existing tuition to that of other schools' and established an appropriate price.
With this increase, St. Benedict's tuition is in the same range as local private schools including Union Catholic Regional High School and Seton Hall Preparatory School. Parents from Newark, who are demographically different than many of the parents in neighboring schools, and are considering sending their sons to St. Benedict's Prep, may be discouraged after seeing this large sum.Without consistently and emphatically conveying this crucial information about financial aid to parents, the school stands to risk losing potential students and endangers its mission of serving the sons of inner-city parents. In fact, one UDII from Newark said that if he had seen the $12,500 price-tag, he would not have applied.
Financial aid will still be available to those students who perform well in their three "A's"—Academics, Attitude, and Activities—and have similar financial circumstances. If students fulfill these reasonable conditions, their contributions towards their tuition are expected to remain at the same rate as prior years; they will receive additional financial aid to compensate for the increase.
This is important information for prospective and current parents and students to fully understand.
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Did FIFA 17 Live Up to its Hype?
Verbal Desensitization: What Does That F***ing Mean?
The "Republican Savior": Marco Antonio Rubio
Witnessing Peace at the United Nations
We Can Fix Cheating If We Can Strengthen the Honor Code Committee First
Why is Music Piracy So High?
The student news site of St. Benedict's Prep in Newark, New Jersey.


Mouser Electronics, Inc., the New Product Introduction (NPI) leader that empowers innovation, today announced a new global distribution partnership with Renesas Electronics Corporation, a leading supplier of advanced semiconductor solutions. Launching the relationship at Embedded World, Mouser and Renesas will focus availability on the latest software, development kits, and microcontrollers (MCUs) that comprise the groundbreaking Renesas Synergy™ Platform. The Synergy Platform is designed to remove obstacles for design engineers and speed time to market for Internet of Things (IoT), industrial, and real-time operating system-based applications.
The Synergy Platform provides developers with immediate access to a complete embedded platform, starting with an application programming interface (API) to leverage a full software framework that is built around the best-in-class Express Logic X-Ware™. This software framework, the Synergy Software Package (SSP), relieves developers from creating and maintaining low-level software repeatedly for each of their embedded projects. The platform also integrates a wide range of scalable ARM®Cortex®-M based MCUs fully accessible through the software APIs, powerful features of the industry-leading development toolchain IAR Embedded Workbench with intuitive configuration assistance, and hardware kits for development and solutions. Since everything has been pre-integrated, tested, and maintained by Renesas, developers can save months of work and extensive investment, freeing them to focus on innovation and differentiation of their end-products.
The Synergy MCU family at Mouser includes four different series of software-, architecture-, and pin-compatible devices. The series are tiered, starting at the high-performance S7 Series with devices operating at 240 MHz, followed by the high-integration S5 Series operating at 120 MHz, the high-efficiency S3 Series at 48 MHz, and the ultra-low-power S1 Series at 32 MHz. These devices implement easy connectivity, rock-sold security, and dependable safety, and facilitate the creation of easy-to-use human-machine interfaces (HMI).
Mouser offers a selection of intuitive Synergy Kits based on S7G2 Group MCUs to simplify embedded software development. The DK-S7G2 development kit provides full access to all MCU pins and features for deeper evaluation and development. TheSK-S7G2 Starter Kit provides a low-cost entry point into the entire Synergy Platform with a touch-enabled color QVGA TFT display. Finally, the PE-HMI1 Product Example kit provides a connected HMI panel with 7-inch WVGA color display plus high-speed wired and wireless communications to serve as a baseline for end-product development.
This entry was posted on Wednesday, March 22nd, 2017 at 1:43 pm	and is filed under Advertiser, News.


When Joseph Stalin dies in 1953, Deputy General Secretary Malenkov is named as acting Premier. However, behind the scenes there is a chaotic struggle for true power, with Khrushchev and Beria in particular vying for the loyalty of their comrades, with often farcical – and sometimes lethal – consequences.
WE SAY: As dark as satire comes – it could be argued that it verges on comedy-horror, as the brilliant ensemble cast stumble their way through a scenario that nobody has dared prepare a contingency for. Iannucci at his brilliant, subversive best.
Doors for this event will open at 13:00. The film will commence at approximately 13:15. The event will conclude at approximately 15:00.
First of two chances to catch the cult favourite - set in 2019! - among the hands-on exhibits at We The Curious.